scholarly journals The contact layer method in the problem of bending of a multilayer beam. Fourier series solution

2017 ◽  
Author(s):  
Andreev Vladimir I. ◽  
Turusov Robert A. ◽  
Tsybin Nikita Yu.
Keyword(s):  
Fourier Series ◽  
Series Solution ◽  
Contact Layer ◽  
Layer Method ◽  
Multilayer Beam

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

A General Solution of Reynolds’ Equation for a Full Finite Journal Bearing

1967 ◽  
Vol 89 (2) ◽  
pp. 203-210 ◽  
Author(s):  
R. R. Donaldson
Keyword(s):  
Fourier Series ◽  
Series Solution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Load Capacity ◽  
Separation Of Variables ◽  
Hill Equation ◽  
Eigenvalues And Eigenvectors ◽  
General Series ◽  
Bearing Load

Reynolds’ equation for a full finite journal bearing lubricated by an incompressible fluid is solved by separation of variables to yield a general series solution. A resulting Hill equation is solved by Fourier series methods, and accurate eigenvalues and eigenvectors are calculated with a digital computer. The finite Sommerfeld problem is solved as an example, and precise values for the bearing load capacity are presented. Comparisons are made with the methods and numerical results of other authors.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

scholarly journals A Fourier Series Solution for Transient Three‐Dimensional Thermohaline Convection in Porous Enclosures

2020 ◽  
Vol 56 (11) ◽  
Author(s):  
Sara Tabrizinejadas ◽  
Marwan Fahs ◽  
Behzad Ataie‐Ashtiani ◽  
Craig T. Simmons ◽  
Raphaël Chiara Roupert ◽  
...  
Keyword(s):  
Fourier Series ◽  
Series Solution ◽  
Three Dimensional ◽  
Thermohaline Convection

scholarly journals Fourier series solution for an anisotropic and layered configuration of the dispersive Henry Problem

2018 ◽  
Vol 54 ◽  
pp. 00014
Author(s):  
B. Koohbor ◽  
M. Fahs ◽  
B. Belfort ◽  
B. Ataie-Ashtiani ◽  
C. T. Simmons
Keyword(s):  
Fourier Series ◽  
Diffusion Coefficients ◽  
Series Solution ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Salt Transport ◽  
Spectral Space ◽  
Finite Element Code ◽  
Henry Problem ◽  
Lower Diffusion

Henry Problem (HP) still plays an important role in benchmarking numerical models of seawater intrusion (SWI) as well as being applied to practical and managerial purposes. The popularity of this problem is due to having a closed-form semi-analytical (SA) solution. The early SA solutions obtained for HP were limited to extensive assumptions that restrict its application in practical works. Several further studies expended the generality of the solution by assuming lower diffusion coefficients or including velocity-dependent dispersion in the results. However, all these studies are limited to homogeneous and isotropic domains. The present work made an attempt to improve the reality of the SA solution obtained for dispersive HP by considering anisotropic and stratified heterogeneous coastal aquifers. The solution is obtained by defining Fourier series for both stream function and salt concentration, applying a Galerkin treatment using the Fourier modes as trial functions and solving the flow and the salt transport equations simultaneously in the spectral space. In order to include stratified heterogeneity, a special depth-hydraulic conductivity model is applied that can be solved analytically without significant mathematical complexity. Several examples are proposed and studied. The results show excellent agreement between the SA and numerical solutions obtained with an in-house advanced finite element code.

Simple Explicit Equations for Transient Heat Conduction in Finite Solids

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
A. G. Ostrogorsky
Keyword(s):  
Fourier Series ◽  
Heat Conduction ◽  
Biot Number ◽  
Series Solution ◽  
Transient Heat Conduction ◽  
Simple Equation ◽  
Lower Error ◽  
Transient Conduction ◽  
The One ◽  
Cylinders And Spheres

Abstract Based on the one-term Fourier series solution, a simple equation is derived for low Biot number transient conduction in plates, cylinders, and spheres. In the 0<Bi<0.3 range, the solution gives approximately three times less error than the lumped capacity solution. For asymptotically low values of Bi, it approaches the lumped capacity solution. A set of equations valid for 0<Bi<1 is developed next. These equations are more involved but give approximately ten times lower error than the lumped capacity solution. Finally, a set of broad-range correlations is presented, covering the 0<Bi<∞ range with less than 1% error.

Modal simulation of unidirectional fluid dynamics (water hammer)

SIMULATION ◽  
1965 ◽  
Vol 5 (3) ◽  
pp. 180-189 ◽  
Author(s):  
Dara Childs
Keyword(s):  
Fluid Dynamics ◽  
Fourier Series ◽  
Differential Equations ◽  
Finite Number ◽  
Series Solution ◽  
Water Hammer ◽  
Simulation Approach ◽  
Frequency Range ◽  
Analog Simulation ◽  
Numerical Examples

This paper presents a general approach to analog simulation of "water-hammer" type phenomena. In brief, the simulation approach consists of obtaining a general Fourier-series time solution in terms of boundary variables (pressures or flowrates), retaining a finite number of terms in this series solution, and then obtaining the ordinary differential equations to which the terms correspond. One customarily "winces" when contemplating a series solution of any form, particularly when informed that the solu tion is to be subsequently approximated by retaining only a finite number of terms. Although the mathe matical manipulations contained in the paper may appear forbidding, they need be developed but once, and the final differential equations which result are quite simple. Truncation of the Fourier series solu tions happily results only in a truncation of the valid frequency range. Within this valid frequency range there is little or no loss in accuracy. These and other advantageous characteristics of the modal simula tion technique are developed by various analytical and numerical examples.

scholarly journals DETERMINATION OF STRESS-STRAIN STATE OF A THREE-LAYER BEAM WITH APPLICATION OF CONTACT LAYER METHOD

Vestnik MGSU ◽  
2016 ◽  
pp. 17-26 ◽  
Author(s):  
Vladimir Igorevich Andreev ◽  
Robert Alekseevich Turusov ◽  
Nikita Yur’evich Tsybin
Keyword(s):  
Experimental Data ◽  
Edge Effect ◽  
Strain State ◽  
Rectangular Cross Section ◽  
Contact Layer ◽  
Theory Of Elasticity ◽  
Stress Strain ◽  
Tangential Stresses ◽  
Stress Strain State ◽  
Layer Method

The article deals with the solution for the stress-strain state of a multilayer composite beam with rectangular cross-section, which is bended by normally distributed load. The intermolecular interaction between layers is accomplished by the contact layer, in which the substances of adhesive and substrate are mixed. We consider the contact layer as a transversal anisotropic medium with such parameters that it can be represented as a set of short elastic rods, which are not connected to each other. For simplicity, we assume that the rods are normally oriented to the contact surface. The contact layer method allows us to solve the problem of determining the concentration of tangential stresses arising at the boundaries between the layers and the corner points, their changes, as well as to determine the physical properties of the contact layer basing on experimental data. Resolving the equations obtained in this article can be used for the solution of many problems of the theory of layered substances. These equations were derived from the fundamental laws of the theory of elasticity and generally accepted hypotheses of the theory of plates for the general case of the bending problem of a multilayer beam with any number of layers. The article deals with the example of the numerical solution of the problem of bending of a three-layer beam. On the basis of this solution the curves were obtained, which reflect the stress-strain state of one of the layers. All these curves have a narrow area of the edge effect. The edge effect is associated with a large gradient tangential stresses in the contact layer. The experimental data suggest that in this zone the destruction of the samples occurs. This fact allows us to say that the equations obtained in this article can be used to construct a theory of the strength layered beams under bending.

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