scholarly journals The inverse problem of recovering an unsteady linear load for an elastic rod of finite length

2020 ◽  
Vol 18 (4) ◽  
pp. 687-692
Author(s):  
Yana Vahterova ◽  
Gregory Fedotenkov
Keyword(s):  
Inverse Problems ◽  
Load Distribution ◽  
Stationary Problem ◽  
Elastic Rod ◽  
System Of Integral Equations ◽  
New Methods ◽  
Linear Load ◽  
Tikhonov Regularisation ◽  
Regularisation Method ◽  
Rod Of Finite Length

The main purpose of the paper is to obtain solutions for new non-stationary inverse problems for elastic rods. The objective of this study is to develop and implement new methods, approaches and algorithms for solving non-stationary inverse problems of rod mechanics. The direct non-stationary problem for an elastic rod consists in determining elastic displacements, which satisfies a given equation of non-stationary oscillations in partial derivatives and some given initial and boundary conditions. The solution of inverse retrospective problems with a completely unknown space-time law of load distribution is based on the method of influence functions. With its application, the inverse retrospective problem is reduced to solving a system of integral equations of the Volterra type of the first kind in time with respect to the sought external axial load of the elastic rod. To solve it, the method of mechanical quadratures is used in combination with the Tikhonov regularisation method.

scholarly journals XI . A mathematical theory of magnetism.—continuation of part I

1851 ◽  
Vol 141 ◽  
pp. 269-285 ◽  
Keyword(s):  
Inverse Problems ◽  
Royal Society ◽  
Mathematical Theory ◽  
Resultant Force ◽  
Normal Section ◽  
New Methods ◽  
Different Parts ◽  
Do So

65. In the course of some researches upon inverse problems regarding distributions of magnetism, and upon the comparison of electro-magnets and common magnets, I have found it extremely convenient to make use of definite terms to express certain distributions of magnetism and forms of magnetized matter possessing remarkable properties. The use of such terms will be of still greater consequence in describing the results of these researches, and therefore, before proceeding to do so, I shall give definitions of the terms which I have adopted, and explain briefly the principal properties of the magnetic distributions to which they are applied. The remainder of this chapter will be devoted to three new methods of analysing the expressions for the resultant force of a magnet at any point, suggested by the consideration of these special forms of magnetic distribution. A Mathematical Theory of Electro-Magnets, and Inverse Problems regarding magnetic distributions, are the subjects of papers which I hope to be able to lay before the Royal Society on a subsequent occasion. 66. Definitions and explanations regarding Magnetic Solenoids (1.) A magnetic solenoid is an infinitely thin bar of any form, longitudinally magnetized with an intensity varying inversely as the area of the normal section in different parts.

scholarly journals GLOBAL SOLUTIONS OF THE EQUATION OF THE KIRCHHOFF ELASTIC ROD IN SPACE FORMS

2012 ◽  
Vol 88 (1) ◽  
pp. 70-80 ◽  
Author(s):  
SATOSHI KAWAKUBO
Keyword(s):  
Space Form ◽  
Global Solutions ◽  
Infinite Length ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Lagrange Equations ◽  
Initial Value ◽  
Space Forms ◽  
Equilibrium Configurations ◽  
Rod Of Finite Length

AbstractThe Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

scholarly journals The inverse non-stationary problem of identification of defects in an elastic rod

2021 ◽  
Vol 13 (S) ◽  
pp. 57-66
Author(s):  
Grigory V. FEDOTENKOV ◽  
Dmitry I. MAKAREVSKII ◽  
Yana A. VAHTEROVA ◽  
Trah Quyet THANG
Keyword(s):  
Solid Mechanics ◽  
Initial Conditions ◽  
Practical Interest ◽  
Stationary Problem ◽  
Technical Condition ◽  
Elastic Rod ◽  
Elastic Bodies ◽  
Fundamental Research ◽  
Great Practical Interest ◽  
Elastic Inclusions

Non-stationary inverse problems of deformed solid mechanics are among the most underexplored due to, inter alia, increasing dimension of non-stationary problems per unit as compared with stationary and static problems, as well as necessity to consider the initial conditions. In the context of the continuing progress of the aviation and aerospace industries, the question arises about technical condition monitoring of aircraft for the purposes of their safe operation. A large proportion of an aircraft structure consists of beam and rod elements exposed to various man-made and natural effects which cause defects inaccessible for visual inspection and required to be identified well in advance. It is well known that defects (such as cracks, cavities, rigid and elastic inclusions) are concentrators of stresses and largely cause processes, which lead to the destruction of elastic bodies. Therefore, the problem of identification of such defects and their parameters, i.e. the problem of identification, represents a great practical interest. Mathematically, the problem of identification represents a non-linear inverse problem. The development of methods of solving such problems is currently a live fundamental research issue.

An Accurate Method for Calculating Load Distribution Factor Kβ of Involute Gears

2007 ◽  
Vol 339 ◽  
pp. 458-462
Author(s):  
D.C. Feng
Keyword(s):  
Axial Force ◽  
Load Distribution ◽  
Gear Tooth ◽  
Accurate Method ◽  
Distribution Factor ◽  
Linear Load ◽  
Tooth Width ◽  
Involute Gears ◽  
Load Distribution Factor ◽  
Gear Manufacture

A new method is given to determine the load distribution factor Kβ of involute gears. Compared with the old method, the new one gives up some improper assumptions such as linear load distribution along gear tooth width. While considering the effects of bearing deformations, gear manufacture and assembly errors, gear axial force and gear tooth run-in on Kβ, the paper calculates Kβ according to “Load- Deformation Coordination” of gear tooth. Practical examples show that this method is more accurate, more effective than old ones.

scholarly journals An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods

2002 ◽  
Vol 2 (8) ◽  
pp. 407-435 ◽  
Author(s):  
Shinuk Kim ◽  
Kevin L. Kreider
Keyword(s):  
Inverse Problems ◽  
Scattering Problem ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Scattering Problems ◽  
Mathematical Basis ◽  
Cross Sectional ◽  
Weakly Nonlinear ◽  
The Time Domain ◽  
Nonlinear Elastic

Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain. The method of wave splitting is employed to formulate a standard scattering problem, forming the mathematical basis for both direct and inverse problems. A quasi-linear version of the Wendroff scheme (FDTD) is used to solve the direct problem. To solve the inverse problem, an asymptotic expansion is used for the wave field; this linearizes the order equations, allowing the use of standard numerical techniques. Analysis and numerical results are presented for three model inverse problems: (i) recovery of the nonlinear parameter in the stress-strain relation for a homogeneous elastic rod, (ii) recovery of the cross-sectional area for a homogeneous elastic rod, (iii) recovery of the elastic modulus for an inhomogeneous elastic rod.

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