Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness

2018 ◽  
Vol 19 (3-4) ◽  
pp. 351-356
Author(s):  
Feng Wang ◽  
Yu-fang Zheng ◽  
Chang-ping Chen
Keyword(s):  
Differential Equations ◽  
Variable Thickness ◽  
Plate Theory ◽  
Nonlinear Differential Equations ◽  
Thin Plates ◽  
Thickness Variation ◽  
Algebraic Equations ◽  
Nonlinear Bending ◽  
Nonlinear Load ◽  
Magnetic Potentials

AbstractWith employing the von Karman plate theory, and considering the linearly thickness variation in one direction, the bending problem of a rectangular magnetoelectroelastic plates with linear variable thickness is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction for the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates with linear variable thickness are established based on the Hamilton’s principle. The Galerkin procedure is taken to translate a set of differential equations into algebraic equations. The numerical examples are presented to discuss the influences of the aspect ratio and span–thickness ratio on the nonlinear load–deflection curves for magnetoelectroelastic plates with linear variable thickness. In addition, the induced electric and magnetic potentials are also presented with the various values of the taper constants.

Nonlinear Responses of Rectangular Magnetoelectroelastic Plates with Transverse Shear Deformation

2016 ◽  
Vol 689 ◽  
pp. 103-107 ◽  
Author(s):  
Yu Fang Zheng ◽  
Tao Chen ◽  
Feng Wang ◽  
Chang Ping Chen
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Plate Theory ◽  
Nonlinear Differential Equations ◽  
Shear Deformation Theory ◽  
Transverse Shear Deformation ◽  
Algebraic Equations ◽  
Nonlinear Load ◽  
Magnetic Potentials

With employing the transverse shear deformation theory and von Karman plate theory, the nonlinear static behavior of a simply supported rectangular magnetoelectroelastic plates is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction of the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates are established based on the Hamilton’s principle. The Galerkin procedure furnishes an infinite system of differential equations into algebraic equations. In the numerical calculations, the effects of the nonlinearity and span-thickness ratio on the nonlinear load-deflection curves and electric/magnetic potentials for magnetoelectroelastic plates are discussed.

scholarly journals Elastic Buckling Behaviour of a Four-Lobed Cross Section Cylindrical Shell with Variable Thickness under Non-Uniform Axial Loads

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Mousa Khalifa Ahmed
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Transfer Matrix ◽  
Cross Section ◽  
Variable Thickness ◽  
Nonlinear Differential Equations ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Thickness Variation ◽  
Matrix Differential

The static buckling of a cylindrical shell of a four-lobed cross section of variable thickness subjected to non-uniform circumferentially compressive loads is investigated based on the thin-shell theory. Modal displacements of the shell can be described by trigonometric functions, and Fourier's approach is used to separate the variables. The governing equations of the shell are reduced to eight first-order differential equations with variable coefficients in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the nonlinear differential equations of the cylindrical shells by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The transfer matrix approach is used to get the critical buckling loads and the buckling deformations for symmetrical and antisymmetrical shells. Computed results indicate the sensitivity of the critical loads and corresponding buckling modes to the thickness variation of cross section and the radius variation at lobed corners of the shell.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Reliable Method ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Implementation Process ◽  
Haar Wavelet ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

scholarly journals Investigating Jeffery-Hamel flow with high magnetic field and nanoparticle by HPM and AGM

2014 ◽  
Vol 4 (4) ◽  
Author(s):  
A. Rostami ◽  
M. Akbari ◽  
D. Ganji ◽  
S. Heydari
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Volume Fraction ◽  
Nonlinear Differential Equations ◽  
Reynolds Numbers ◽  
Solution Procedure ◽  
Algebraic Equations ◽  
Divergent Channel ◽  
Mathematical Operations

AbstractIn this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using two powerful analytical methods, Homotopy Perturbation Method (HPM) and a simple and innovative approach which we have named it Akbari-Ganji’s Method(AGM). Comparisons have been made between HPM, AGM and Numerical Method and the acquired results show that these methods have high accuracy for different values of α, Hartmann numbers, and Reynolds numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.It is necessary to represent some of the advantages of choosing the new method, AGM, for solving nonlinear differential equations as follows: AGM is a very suitable computational process and is applicable for solving various nonlinear differential equations. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration, the solution of the problem can be obtained very simply and easily. It is notable that this solution procedure, AGM, can help students with intermediate mathematical knowledge to solve a broad range of complicated nonlinear differential equations.

Buckling Analysis of Circular FGM Plate Having Variable Thickness Under Uniform Compression by Finite Element Method

2005 ◽  
Author(s):  
Abazar Shamekhi ◽  
Mohammad H. Naei
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variable Thickness ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Buckling Load ◽  
Thin Plates ◽  
Thickness Variation ◽  
Simply Supported ◽  
Element Method

This study presents the buckling analysis of radially-loaded circular plate with variable thickness made of functionally-graded material. The boundary conditions of the plate is either simply supported or clamped. The stability equations were obtained using energy method based on Love-Kichhoff hypothesis and Sander’s non-linear strain-displacement relation for thin plates. The finite element method is used to determine the critical buckling load. The results obtained show good agreement with known analytical and numerical data. The effects of thickness variation and Poisson’s ratio are investigated by calculating the buckling load. These effects are found not to be the same for simply supported and clamped plates.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

2019 ◽  
Vol 17 (04) ◽  
pp. 1950026 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Type Equation ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
Quasilinearization Technique ◽  
Distribution Equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

Angular Rate From the Nine Accelerometer Instrument

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Jan P. B. Vreeburg
Keyword(s):  
Differential Equations ◽  
Real Time ◽  
Quadratic Forms ◽  
Multiple Solutions ◽  
Nonlinear Differential Equations ◽  
Angular Rate ◽  
Multiple Roots ◽  
Algebraic Equations ◽  
Real Time Processing ◽  
Time Processing

Instruments composed of arrangements of linear accelerometers are used to determine motion variables. With nine accelerometers, the angular rate of the instrument can be resolved algebraically rather than by the solution of a system of nonlinear differential equations. The algebraic equations consist of three quadratic forms in the angular rate and, in general, lead to multiple solutions. The number and distribution of the multiple roots have been analyzed. Examples illustrate the application of the analyses, for batch and real-time processing. Suitable algorithms have been developed and are detailed. Subjects for further study have been indicated.

scholarly journals Multiscale homogenization in Kirchhoff's nonlinear plate theory

2015 ◽  
Vol 25 (09) ◽  
pp. 1765-1812 ◽  
Author(s):  
Laura Bufford ◽  
Elisa Davoli ◽  
Irene Fonseca
Keyword(s):  
Dimension Reduction ◽  
Plate Theory ◽  
Thin Plates ◽  
Nonlinear Bending ◽  
Nonlinear Plate ◽  
Relative Ratio ◽  
Nonlinear Elastic ◽  
Multiscale Homogenization ◽  
Thickness Parameter ◽  
Limit Models

The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff's nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter h and the two homogenization scales ε and ε2.

Composite Circular Plates With Residual Tensile Stress Undergoing Large Deflections

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
Brian Homeijer ◽  
Benjamin A. Griffin ◽  
Matthew D. Williams ◽  
Bhavani V. Sankar ◽  
Mark Sheplak
Keyword(s):  
Tensile Stress ◽  
Residual Stresses ◽  
Composite Plate ◽  
Plate Theory ◽  
Nonlinear Differential Equations ◽  
High Stress ◽  
Thin Plates ◽  
Residual Tensile Stress ◽  
Mechanical Sensitivity ◽  
Element Analysis

Many micromachined electroacoustic devices use thin plates in conjunction with electrical components to measure acoustic signals. Composite layers are needed for electrical passivation, moisture barriers, etc. The layers often contain residual stresses introduced during the fabrication process. Accurate models of the composite plate mechanics are crucial for predicting and optimizing device performance. In this paper, the von Kármán plate theory is implemented for a transversely isotropic, axisymmetric plate with in-plane tensile stress and uniform transverse pressure loading. A numerical solution of the coupled force-displacement nonlinear differential equations is found using an iterative technique. The results are verified using finite element analysis. This paper contains a study of the effects of tensile residual stresses on the displacement field and examines the transition between linear and nonlinear behavior. The results demonstrate that stress stiffening in the composite plate delays the onset of nonlinear deflections and decreases the mechanical sensitivity. In addition, under high stress the plate behavior transitions to that of a membrane and becomes insensitive to the composite nature of the plate. The results suggest a tradeoff between mechanical sensitivity and linearity.

scholarly journals Piecewise continuous approach to nonlinear differential equations approximation problem of computational structural mechanics

2018 ◽  
Vol 251 ◽  
pp. 04024
Author(s):  
Roman Leibov
Keyword(s):  
Differential Equations ◽  
Random Search ◽  
Piecewise Linear ◽  
Nonlinear Differential Equations ◽  
Piecewise Linear Approximation ◽  
Continuous Approximation ◽  
Algebraic Equations ◽  
Computational Structural Mechanics ◽  
Piecewise Continuous ◽  
Linear Differential

This paper presents a nonlinear differential equations system piecewise continuous approximation. The piecewise continuous approximation improves piecewise linear approximation through reducing the errors at the boundaries of different linear differential equations systems areas. The matrices of piecewise continuous differential and algebraic equations systems are estimated using nonlinear differential equations system time responses and random search method. The results of proposed approach application are presented.

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