scholarly journals An Elastodiffusive Orthotropic Euler–Bernoulli Beam Considering Diffusion Flux Relaxation

2019 ◽  
Vol 24 (1) ◽  
pp. 23 ◽  
Author(s):  
Dmitry Tarlakovskii ◽  
Andrei Zemskov
Keyword(s):  
Fourier Series ◽  
Variational Principle ◽  
Laplace Transform ◽  
Diffusion Flux ◽  
Fourier Series Expansion ◽  
Bernoulli Beam ◽  
Elastic Diffusion ◽  
The Laplace Transform ◽  
Flux Relaxation ◽  
Euler Bernoulli Beam

This article considers an unsteady elastic diffusion model of Euler–Bernoulli beam oscillations in the presence of diffusion flux relaxation. We used the model of coupled elastic diffusion for a homogeneous orthotropic multicomponent continuum to formulate the problem. A model of unsteady bending for the elastic diffusive Euler–Bernoulli beam was obtained using Hamilton’s variational principle. The Laplace transform on time and the Fourier series expansion by the spatial coordinate were used to solve the obtained problem.

The Response of Infinite Strings and Beams to an Initially Applied Moving Force: Analytical Solution

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
André Langlet ◽  
Ophélie Safont ◽  
Jérôme Renard
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Stationary Solution ◽  
Timoshenko Beam ◽  
Bernoulli Beam ◽  
Moving Force ◽  
The Laplace Transform ◽  
Bar Velocity ◽  
Infinite Strings ◽  
Euler Bernoulli Beam

This paper presents the analytical solutions for bilaterally infinite strings and infinite beams on which a point force is initially applied, which then moves on the structure at a constant velocity. The solutions are sought by first applying the Fourier transform to the spatial coordinate dependence, and then the Laplace transform to the time variable of dependence, of the governing equations of motion. For the strings, it is necessary to distinguish between the case of a sonic load (a force moving at the phase velocity of transverse waves) and the cases of subsonic and supersonic loads. This is achieved by a suitable expansion in polynomial ratios of the Laplace transform, before going back to the original Fourier transform, whose inverse is obtained by exact calculations of the integrals over the complex infinite domain. For the Euler-Bernoulli beam, the same process leads to the closed-form (exact) formula for the displacement, from which the stress can be deduced. The displacement consists of the sum of two integrals: one representing the transient part, and the other, the stationary part of the solution. The stationary part is observed in the vicinity of the force for a very long travel time. The transient part is observed at a finite position coordinate, in relative proximity to the starting point of the moving force. For the Timoshenko beam, the final step in the calculation of the displacement and rotation, which requires a numerical evaluation of the integrals, leads to Fourier cosine and sine transforms. The response of the beam depends on the load velocity, relative to the two characteristic velocities: those of shear waves and longitudinal waves. This demonstrates that the transient parts of the solutions, in the Euler-Bernoulli beam or in the Timoshenko beam, are quasi identical. However, classical theory fails to forecast high frequency responses, occurring with velocities of the load exceeding twenty per cent of the bar velocity. For a velocity greater than the velocity of the shear waves, classical theory wrongly forecasts the response. In addition, according to the Euler-Bernoulli beam theory, the flexural waves are able to exceed the bar velocity, which is not realistic. If the load moves for a long period, the solution in the vicinity of the load tends towards a stationary solution. It is important to note that the solution to the stationary problem must be completed by the solution to the associated homogeneous system to represent the physical stationary solution.

scholarly journals Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Ratchata Theinchai ◽  
Siriwan Chankan ◽  
Weera Yukunthorn
Keyword(s):  
Laplace Transform ◽  
Cantilever Beam ◽  
Adomian Decomposition Method ◽  
Small Deformation ◽  
Approximate Solutions ◽  
Beam Equation ◽  
Bernoulli Beam ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.

scholarly journals The noise handling properties of the Talbot algorithm for numerically inverting the Laplace transform

2018 ◽  
Vol 13 ◽  
pp. 174830181879706 ◽  
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Noisy Data ◽  
Standard Test ◽  
Numerical Inversion ◽  
Test Functions ◽  
Inversion Algorithm ◽  
Inverse Transform ◽  
Exact Data ◽  
The Laplace Transform

This paper examines the noise handling properties of three of the most widely used algorithms for numerically inverting the Laplace transform. After examining the genesis of the algorithms, their error handling properties are evaluated through a series of standard test functions in which noise is added to the inverse transform. Comparisons are then made with the exact data. Our main finding is that the for “noisy data”, the Talbot inversion algorithm performs with greater accuracy when compared to the Fourier series and Stehfest numerical inversion schemes as they are outlined in this paper.

scholarly journals Modeling of vibration for functionally graded beams

2016 ◽  
Vol 14 (1) ◽  
pp. 661-672 ◽  
Author(s):  
Gülsemay Yiğit ◽  
Ali Şahin ◽  
Mustafa Bayram
Keyword(s):  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Fourier Series Expansion ◽  
Bernoulli Beam ◽  
Adomian Decomposition ◽  
Graded Material ◽  
Euler Bernoulli Beam

AbstractIn this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a certain rate. This mixture at a certain rate is expressed with an exponential function in order to try to minimize singularities from transition between different surfaces of materials as much as possible. According to the structure of the ADM in terms of initial conditions of the problem, a Fourier series expansion method is used along with the ADM for the solution of simply supported functionally graded Euler-Bernoulli beams. Finally, by choosing an appropriate mixture rate for the material, the results are shown in figures and compared with those of a standard (homogeneous) Euler-Bernoulli beam.

scholarly journals MODELING OF UNSTEADY COUPLED MECHANODIFFUSION PROCESSES IN A CONTINUUM ISOTROPIC CYLINDER

2020 ◽  
Vol 82 (2) ◽  
pp. 156-167
Author(s):  
N.A. Zverev ◽  
A.V. Zemskov ◽  
D.V. Tarlakovskii
Keyword(s):  
Laplace Transform ◽  
Green's Functions ◽  
Green’S Functions ◽  
Fourier Series Expansion ◽  
Polar Coordinate System ◽  
Problem Solution ◽  
Isotropic Cylinder ◽  
Linear Algebraic Equations ◽  
The Laplace Transform ◽  
And Diffusion

We considered the one-dimensional polar-symmetric problem of stress-strain state determining of a continuum isotropic multicomponent cylinder. The cylinder is affected by unsteady surface elastic diffusive perturbations. The coupled system of elastic diffusion equations in the polar coordinate system is used as a mathematical model. The problem solution is sought in the integral form and is represented as convolutions of Green's functions with functions defining surface elastodiffusive perturbations. Mechanical loads and diffusion fields are considered as external influences. We used the Laplace transform by time, and Fourier series expansion in first kind Bessel functions to find the Green's functions. To calculate the coefficients of these series, we obtained formulas for transforming differential operators of the first, second, and third orders using the Hankel integral transform on a segment, which allowed us to reduce the initial boundary-value problem of mechanodiffusion to a system of linear algebraic equations. Laplace transforms of Green's functions are represented through rational functions of the Laplace transform parameter. The Laplace transform inversion is done analytically due to residues and operational calculus tables. As a result, Analytical expressions of surface Green's functions are obtained for the considering problem. Numerical study of the mechanical and diffusion fields interaction in a continuous isotropic cylinder is performed. We used two-component material as an example. The cylinder is under pressure uniformly distributed over the surface. The solution is presented in analytical form and in the form of three-dimensional graphs of the desired displacement fields and concentration increments as functions of time and radial coordinate. This calculation example allows us to demonstrate the coupling effect of mechanical and diffusion fields. It manifests itself as a change in the concentrations of the continuum components under the influence of external unsteady surface pressure.

scholarly journals Euler-Bernoulli cantilever beam bending considering the inner diffusion flows finite propagation speed

2020 ◽  
pp. 107-114
Author(s):  
Андрей Владимирович Земсков ◽  
Георгий Михайлович Файкин
Keyword(s):  
Mass Transfer ◽  
Propagation Speed ◽  
Bernoulli Beam ◽  
Elastic Diffusion ◽  
Multicomponent Media ◽  
Equivalent Boundary ◽  
Beam Bending ◽  
Finite Propagation Speed ◽  
Euler Bernoulli Beam ◽  
Diffusion Flows

Исследуются нестационарные колебания балки Эйлера-Бернулли с учетом массопереноса. Используется модель упругой диффузии для многокомпонентных сред. Для получения решения задачи используются вариационный принцип Даламбера и метод эквивалентный граничных условий. Unsteady vibrations of the Euler-Bernoulli beam are studied taking into account mass transfer. The model of elastic diffusion for multicomponent media is used. To obtain a solution to the problem, the d’Alembert variational principle and the equivalent boundary conditions method are used.

scholarly journals Fourier Series Approach for the Vibration of Euler–Bernoulli Beam under Moving Distributed Force: Application to Train Gust

2019 ◽  
Vol 2019 ◽  
pp. 1-21
Author(s):  
Shupeng Wang ◽  
Weigang Zhao ◽  
Guangyuan Zhang ◽  
Feng Li ◽  
Yanliang Du
Keyword(s):  
Fourier Series ◽  
Dynamic Response ◽  
High Speed ◽  
Dynamic Pressure ◽  
Superposition Method ◽  
Bernoulli Beam ◽  
Complex Procedure ◽  
A Site ◽  
Site Test ◽  
Euler Bernoulli Beam

The dynamic response of an Euler–Bernoulli beam under moving distributed force is studied. By decomposing the distributed force into Fourier series and extending them to semi-infinite sine waves, the complex procedure for solving this problem is simplified to three base models, which are calculated by the modal superposition method further. The method is proved to be highly accurate and computational efficient by comparing with the finite element method. For verifying the theory and exploring the relationship between dynamic pressure due to train gust and vibration of the structure, a site test was conducted on a platform canopy located on the Beijing-Shanghai high-speed railway in China. The results show the theory can be used to evaluate the dynamic response of the beam structure along the trackside due to the train gust. The dynamic behavior of a 4-span continuous steel purlin is studied when the structure is subjected to the moving pressure due to different high-speed train passing.

Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

2011 ◽  
Vol 189-193 ◽  
pp. 3391-3394 ◽  
Author(s):  
Qing Zhao Yao ◽  
Lin Chao Liu ◽  
Qi Fang Yan
Keyword(s):  
Fractional Derivative ◽  
Mechanical Behavior ◽  
Constitutive Relations ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Bernoulli Beam ◽  
The Laplace Transform ◽  
Static Mechanical ◽  
Euler Bernoulli Beam

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

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