Forced Flexural Vibrations in a Thin Nonlocal Rectangular Plate with Kirchhoff’s Thin Plate Theory

2020 ◽  
Vol 20 (09) ◽  
pp. 2050107
Author(s):  
Iqbal Kaur ◽  
Parveen Lata ◽  
Kulvinder Singh
Keyword(s):  
Rectangular Plate ◽  
Thin Plate ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Generalized Thermoelasticity ◽  
Transversely Isotropic ◽  
Small Scale ◽  
Concentrated Load ◽  
Thin Rectangular Plate ◽  
Flexural Vibrations

This study deals with a novel model of forced flexural vibrations in a transversely isotropic thermoelastic thin rectangular plate (TRP) due to time harmonic concentrated load. The mathematical model is prepared for the thin plate in a closed form with the application of Kirchhoff’s love plate theory for nonlocal generalized thermoelasticity with Green–Naghdi (GN)-III theory of thermoelasticity. The nonlocal thin plate has a nonlocal parameter to depict small-scale effect. The double finite Fourier transform technique has been used to find the expressions for lateral deflection, thermal moment and temperature distribution for simply supported (SS) thin rectangular plate in the transformed domain. The effect of classical thermoelasticity (CTE) theory of thermoelasticity and nonlocal parameters has been shown on the computed quantities. Few particular cases have also been deduced.

scholarly journals Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Phase Portraits ◽  
Small Scale ◽  
Governing System ◽  
Account Due

AbstractIn this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

Experimental studies on the dynamic response of thin rectangular plates subjected to moving mass

2020 ◽  
pp. 107754632093313 ◽  
Author(s):  
Sajjad Seifoori ◽  
Ahmad Mahdian Parrany ◽  
Sajjad Darvishinia
Keyword(s):  
Dynamic Response ◽  
Rectangular Plate ◽  
Plate Theory ◽  
Experimental Studies ◽  
Time History ◽  
Moving Mass ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Thin Rectangular Plate ◽  
Expansion Technique

This article presents experimental studies on the dynamic response of a thin rectangular plate with clamped boundary conditions subjected to a moving mass. The designed experimental setup is described in detail, and the obtained experimental results are compared with theoretical solutions. In this regard, the governing motion equation of the thin rectangular plate excited by a moving mass is formulated based on the classical plate theory, and the eigenfunction expansion technique is used to solve the equation. Parametric studies are carried out to investigate the effect of some parameters, including the moving object mass and velocity, as well as the plate’s aspect ratio and thickness, on the dynamic response of the plate based on the time history of the plate’s central point deflection.

scholarly journals The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1128 ◽  
Author(s):  
Ahmed E. Abouelregal ◽  
Marin Marin
Keyword(s):  
External Force ◽  
Nonlocal Parameter ◽  
Generalized Thermoelasticity ◽  
Nonlocal Elasticity Theory ◽  
Sine Wave ◽  
Harmonic Excitation ◽  
Angular Frequency ◽  
Small Scale ◽  
Length Scale ◽  
Wave Heating

In this article, a nonlocal thermoelastic model that illustrates the vibrations of nanobeams is introduced. Based on the nonlocal elasticity theory proposed by Eringen and generalized thermoelasticity, the equations that govern the nonlocal nanobeams are derived. The structure of the nanobeam is under a harmonic external force and temperature change in the form of rectified sine wave heating. The nonlocal model includes the nonlocal parameter (length-scale) that can have the effect of the small-scale. Utilizing the technique of Laplace transform, the analytical expressions for the studied fields are reached. The effects of angular frequency and nonlocal parameters, as well as the external excitation on the response of the nanobeam are carefully examined. It is found that length-scale and external force have significant effects on the variation of the distributions of the physical variables. Some of the obtained numerical results are compared with the known literature, in which they are well proven. It is hoped that the obtained results will be valuable in micro/nano electro-mechanical systems, especially in the manufacture and design of actuators and electro-elastic sensors.

Closed Form Expression for the Vibration Problem of a Transversely Isotropic Magneto-Electro-Elastic Plate

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Mei-Feng Liu ◽  
Tai-Ping Chang
Keyword(s):  
Closed Form ◽  
Thin Plate ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Plate Theory ◽  
Transversely Isotropic ◽  
Closed Form Expression ◽  
Transverse Displacement ◽  
Form Expression ◽  
Thin Plate Theory

A closed form expression for the transverse vibration of a magnetoelectroelastic (MEE) thin plate is derived, and the exact solution for the free vibration of a two-layered BaTiO3–CoFe2O4 composite is obtained. Based on the Kirchhoff thin plate theory, the bending problem of a transversely isotropic MEE rectangular plate is investigated, and the governing equation in terms of the transverse displacement is then presented in a rather compact form. The material coefficients for such MEE plate are expressed uniquely by the volume fraction (vf) of the two-layered BaTiO3–CoFe2O4 composite, which indicates a transversely isotropic MEE medium. The natural frequencies of such MEE plate are evaluated analytically, and the effects of different volume fractions on the natural frequency are further discussed.

Moving loads on ice plates of finite thickness

1991 ◽  
Vol 226 ◽  
pp. 37-61 ◽  
Author(s):  
J. Strathdee ◽  
W. H. Robinson ◽  
E. M. Haines
Keyword(s):  
Thin Plate ◽  
Green's Functions ◽  
Green’S Functions ◽  
Plate Theory ◽  
Moving Load ◽  
Finite Thickness ◽  
Plate Thickness ◽  
Concentrated Load ◽  
Model Calculations ◽  
Thin Plate Theory

The response of a floating ice plate to a moving load is given in terms of a pair of Green's functions. General expressions for these Green's functions are derived for the case of an infinite isotropic plate of uniform thickness supported on a fluid base of uniform depth. The distributions of stress and strain in the vicinity of a concentrated load receive significant contributions from waves of length comparable with the plate thickness and their description necessitates an exact description of thickness effects. Circumstances in which the classical thin-plate theory can be recovered are discussed. The steady-state response to a uniformly moving load displays a so-called ‘critical’ behaviour for load velocities in the neighbourhood of a threshold value at which radiation commences. At the critical speed the amplitude is limited by dissipative forces in the ice plate. To describe this a simple viscoelastic term is included in our model. Calculations indicate that thin-plate theory is accurate to within 5% for distances greater than twenty times the ice thickness.

A Free Boundary Value Problem in Plate Theory

1983 ◽  
Vol 50 (2) ◽  
pp. 297-302 ◽  
Author(s):  
P. Villaggio
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Rectangular Plate ◽  
Elastic Foundation ◽  
Equilibrium Configuration ◽  
Plate Theory ◽  
Contact Region ◽  
Free Boundary Value Problem ◽  
Concentrated Load ◽  
Winkler Model

In this paper the equilibrium configuration of a thin rectangular plate, supported on an elastic foundation that reacts in compression only, is studied. It is assumed that the foundation is described by the Winkler model in the contact region. The plate is supposed to be weightless and subjected to a concentrated load in its midpoint. The shape of the free boundary, where the plate loses contact with the foundation, is determined.

Experimental Investigation of Nonlinear Vibration of a Thin Rectangular Plate

2019 ◽  
Vol 11 (06) ◽  
pp. 1950059 ◽  
Author(s):  
Sohayb Abdulkerim ◽  
Athanasios Dafnis ◽  
Hans-G Riemerdes
Keyword(s):  
Rectangular Plate ◽  
Plate Theory ◽  
Plate Thickness ◽  
Extraction Procedure ◽  
Time History ◽  
Laser Vibrometer ◽  
Classical Plate Theory ◽  
Thin Rectangular Plate ◽  
Sinusoidal Input ◽  
History Of

In this paper, the geometrical nonlinear vibrations of a rectangular plate have been investigated experimentally and numerically. The experiment was conducted on a thin rectangular plate. The plate was excited close to the first fundamental natural frequency. The time history of velocities of the central point has been measured by using a laser vibrometer. While the numerical investigation has been carried using the Finite Element Method (FEM), the numerical results are validated by analytical and experimental results. In order to develop and test the extraction procedure of the bifurcation plot of a dynamical system, a chaotic pendulum has been analyzed. Then, the same successful code has been used again for the experimental dynamics of the investigated plate. The plate has been forced with a sinusoidal input at a gradually stepped and increased amplitude. For every step, the phase portrait is determined, and then processed to extract the bifurcation map. The resulted map has shown successfully the linear range where the classical plate theory is adequate, and the boundary at which the transition to nonlinearity has occurred. The bifurcation has occurred when the lateral amplitude has reached 50% of the plate thickness.

The Elastically Clamped Rectangular Plate With Arbitrary Lateral and Thermal Loading

1962 ◽  
Vol 29 (3) ◽  
pp. 578-580
Author(s):  
C. C. Chao ◽  
Max Anuliker
Keyword(s):  
Rectangular Plate ◽  
Thin Plate ◽  
Infinite Series ◽  
Series Solution ◽  
Thermal Loading ◽  
Plate Theory ◽  
Convergent Series ◽  
Clamped Plate ◽  
Simply Supported ◽  
Thin Plate Theory

Within the limits of classical thin-plate theory a variety of elementary problems have been solved for the rectangular plate3,4,5. In particular, the rectangular plate with edges simply supported or clamped has been dealt with at length and the solution to different loading cases given either in the form of a doubly infinite series or a single infinite series. In this paper a rapidly convergent series solution is outlined for the uniformly elastically clamped plate which is subjected to nonuniform lateral and thermal loading. The solution converges in the limit to those corresponding to the simply supported and rigidly clamped plate.

scholarly journals Modelling Method of Dynamic Characteristics of Marine Thin-Walled Structure

2019 ◽  
Vol 26 (4) ◽  
pp. 39-46
Author(s):  
Do Van Doan ◽  
Adam Szeleziński ◽  
Lech Murawski ◽  
Adam Muc
Keyword(s):  
Numerical Model ◽  
Rectangular Plate ◽  
Thin Plate ◽  
Vibration Analysis ◽  
Three Dimensional ◽  
Rectangular Thin Plate ◽  
Thin Rectangular Plate ◽  
Thin Walled Structures ◽  
Modelling Method ◽  
Thin Walled

AbstractThin-walled structures are very popular in industries, especially in the field of shipbuilding. There are many types of equipment and structures of ships, which are made up of thin-walled structures such as hull, deck and superstructure. Therefore, the analysis and understanding of the static and dynamic characteristics of a thin-walled structure are very important. In this article, we focus on vibration analysis of a typical thin-walled structure-rectangular plate, a basic structure of the hull. Vibration analysis of a rectangular thin plate is conducted by two methods: numerical modelling method of the finite element on Patran-Nastran software platform and experimental method implemented in the laboratory of Gdynia Maritime University. Thin rectangular plate is fixed one end by four clamping plates and is modelled with finite elements and different meshing densities. The numerical model of thin rectangular plate is divided into four cases. Case 1, thin rectangular plate, and clamping plates are modelled with two-dimensional elements. Case 2, the rectangular thin plate is modelled with two-dimensional elements; the clamping plates are modelled with three-dimensional elements. Case 3, both the rectangular thin plate and clamping plates are modelled with three-dimensional elements. Case 4, the rectangular thin plate, and clamping plates are modelled with three-dimensional elements with larger mesh density to increase the accuracy of the calculation results. After that, the results of vibration analysis according to the numerical modelling method on Patran-Nastran software platform for these cases were compared with the measurement results. From there, assess the accuracy of analysis results of selected numerical model methods and the ability to widely apply this numerical model method to other marine structures.

Scattering from a Thin Rectangular Plate at Glancing Incidence

2001 ◽  
Vol 21 (2) ◽  
pp. 147-163 ◽  
Author(s):  
Hirohide Serizawa ◽  
Kohei Hongo ◽  
Hirokazu Kobayashi
Keyword(s):  
Rectangular Plate ◽  
Thin Rectangular Plate
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