Analytical Solution to the Boundary Value Problem of Steady Creep of a Nonaxisymmetric Thick-Walled Tube under the Action of Internal Pressure

In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.

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Analytical Solution for the Boundary Value Problem of Elastic Beams Subjected to Distributed Load

Journal of Engineering Science ◽

10.21315/jes2021.17.1.5 ◽

2021 ◽

Vol 17 (1) ◽

pp. 75-93

Author(s):

Mustapha Adewale Usman ◽

Nur Nabilah Afja Mohd Afandi ◽

Fatai Akangbe Hammed ◽

Debora Oluwatobi Daniel

Keyword(s):

Differential Equation ◽

Analytical Solution ◽

Boundary Value Problem ◽

Moving Load ◽

Boundary Value ◽

Elastic Beam ◽

Beam Deflection ◽

Elastic Beams ◽

Distributed Load ◽

Order Four

Analytical solution for the boundary value problem (BVP) of elastic beams subjected to distributed load was investigated. Based on the study, dynamic application curves are developed for beam deflection. The partial differential equation of order four were analysed to determine the dynamic response of the elastic beam under consideration and solved analytically. Effects of different parameters such as the mass of the load, the length of the moving load, the distance covered by the moving load, the speed of the moving and the axial force were considered. Result revealed that the values of the deflection with acceleration being considered are higher than the system where acceleration of the moving load is negligible. These obtained results are in agreement with the existing results.

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Analytical Solution of Forced Vibration of a Column Subjected to Conservative and Nonconservative Forces

Volume 6B: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21562 ◽

2001 ◽

Author(s):

Zhixiang Xu ◽

Kunisato Seto ◽

Hideyuki Tamura

Keyword(s):

Analytical Solution ◽

Boundary Value Problem ◽

Integral Transform ◽

Boundary Value ◽

Suspension Bridge ◽

Follower Force ◽

Analysis Process ◽

Finite Integral ◽

Excitation Force ◽

Integral Transform Technique

Abstract This paper presents analytical results of forced transverse vibration of a column with a mass attached at free-end subjected to a tangential follower force and a transverse distributed excitation force, that is a simplified model of some structures in civil and mechanical engineering, e.g., a column of a suspension bridge, a launched rocket in the atmosphere. Because the tangential follower force is nonconservative, it is very difficult to get the analytical solution of the problem by usually-used analysis methods with which the adjoint boundary value problem can not be directly obtained. However, by applying the finite integral transform technique, we directly obtained the adjoint boundary value problem in the analysis process, and successfully obtained the analytical solution of the column’s vibration excited by the transverse distributed force.

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