scholarly journals Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system

2021 ◽  
Vol 5 (7 (113)) ◽  
pp. 14-24
Author(s):  
Valeriy Chigirinsky ◽  
Olena Naumenko
Keyword(s):  
Coordinate System ◽  
Continuum Mechanics ◽  
Theory Of Elasticity ◽  
Applied Problem ◽  
Classical Solutions ◽  
Cartesian Coordinate System ◽  
Beam Shape ◽  
Edge Conditions ◽  
Complex Boundary ◽  
Cartesian Coordinate

Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form. However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors. Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown. By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loading

Schwarzschild solution from Weyl transverse gravity

2017 ◽  
Vol 32 (03) ◽  
pp. 1750022 ◽  
Author(s):  
Ichiro Oda
Keyword(s):  
General Relativity ◽  
Coordinate System ◽  
Conformal Transformation ◽  
Classical Solutions ◽  
Cartesian Coordinate System ◽  
Spacetime Dimension ◽  
Schwarzschild Solution ◽  
Classical Level ◽  
Cartesian Coordinate ◽  
Volume Preserving

We study classical solutions in the Weyl-transverse (WTDiff) gravity. The WTDiff gravity is invariant under both the local Weyl (conformal) transformation and the volume preserving diffeomorphisms (Diff) (transverse diffeomorphisms (TDiff)) and is known to be equivalent to general relativity at least at the classical level. In particular, we find that in a general spacetime dimension, the Schwarzschild metric is a classical solution in the WTDiff gravity when it is expressed in the Cartesian coordinate system.

scholarly journals Reissner–Nordström solution from Weyl transverse gravity

2016 ◽  
Vol 31 (36) ◽  
pp. 1650206 ◽  
Author(s):  
Ichiro Oda
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Electromagnetic Field ◽  
Coordinate System ◽  
Conformal Transformation ◽  
Classical Solutions ◽  
Cartesian Coordinate System ◽  
Classical Level ◽  
Cartesian Coordinate ◽  
Volume Preserving

We study classical solutions in the Weyl-transverse (WTDiff) gravity coupled to an electromagnetic field in four spacetime dimensions. The WTDiff gravity is invariant under both the local Weyl (conformal) transformation and the volume preserving diffeomorphisms (transverse diffeomorphisms) and is known to be equivalent to general relativity at least at the classical level (perhaps even in the quantum regime). In particular, we find that only in four spacetime dimensions, the charged Reissner–Nordström black hole metric is a classical solution when it is expressed in the Cartesian coordinate system.

On the Flexural-Torsional Behavior of a Straight Elastic Beam Subject to Terminal Moments

1993 ◽  
Vol 60 (2) ◽  
pp. 498-505 ◽  
Author(s):  
Z. Tan ◽  
J. A. Witz
Keyword(s):  
Coordinate System ◽  
Cross Section ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Beam ◽  
Cartesian Coordinate System ◽  
Kinematic Constraints ◽  
Torsional Behavior ◽  
Cartesian Coordinate

This paper discusses the large-displacement flexural-torsional behavior of a straight elastic beam with uniform circular cross-section subject to arbitrary terminal bending and twisting moments. The beam is assumed to be free from any kinematic constraints at both ends. The equilibrium equation is solved analytically with the full expression for curvature to obtain the deformed configuration in a three-dimensional Cartesian coordinate system. The results show the influence of the terminal moments on the beam’s deflected configuration.

scholarly journals A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate

2018 ◽  
Vol 1 (4) ◽  
Author(s):  
Debabrata Datta ◽  
T K Pal
Keyword(s):  
Diffusion Equation ◽  
Coordinate System ◽  
Lattice Boltzmann ◽  
Spherical Coordinate System ◽  
Cartesian Coordinate System ◽  
Spherical Coordinate ◽  
Cartesian Coordinate ◽  
Lattice Boltzmann Models ◽  
Boltzmann Method ◽  
Good Agreement

Lattice Boltzmann models for diffusion equation are generally in Cartesian coordinate system. Very few researchers have attempted to solve diffusion equation in spherical coordinate system. In the lattice Boltzmann based diffusion model in spherical coordinate system extra term, which is due to variation of surface area along radial direction, is modeled as source term. In this study diffusion equation in spherical coordinate system is first converted to diffusion equation which is similar to that in Cartesian coordinate system by using proper variable. The diffusion equation is then solved using standard lattice Boltzmann method. The results obtained for the new variable are again converted to the actual variable. The numerical scheme is verified by comparing the results of the simulation study with analytical solution. A good agreement between the two results is established.

Geometrical Optics

2019 ◽  
pp. 188-214
Author(s):  
B. D. Guenther
Keyword(s):  
Coordinate System ◽  
Ray Tracing ◽  
Geometrical Optics ◽  
Paraxial Approximation ◽  
Cartesian Coordinate System ◽  
Lens System ◽  
Thin Lens ◽  
Aperture Stop ◽  
Sign Convention ◽  
Cartesian Coordinate

Discuss the limits imposed by the paraxial approximation. Define the sign convention based on the cartesian coordinate system, the foiundation of analytic geometery. Demonstrate ray tracing technique to derive the ABCD maxtrix which will generate both the gaussian and Newtonian form of the thin lens equation and the lens maker’s equation. The cardinal points of a lens are also derived. The ABCD matrix is used to explore the methods used in ray tracing to locate the aperture stop of a Cooke’s triplet lens system. In the problem set, the student is asked to use the aperture stop to locate the entrance and exit pupil of a Cooke’s triplet.

scholarly journals Approximate Inertial Coordinate System Selections for Rotation Problems - The Gravitational Field of the Celestial Body Higher than the Object Being Rotated

2015 ◽  
Vol 8 (1) ◽  
pp. 102
Author(s):  
Zifeng Li
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Celestial Body ◽  
Angular Speed ◽  
Calculation Error ◽  
Cartesian Coordinate System ◽  
Inertial Coordinate System ◽  
The Earth ◽  
The Galaxy ◽  
Cartesian Coordinate

<p class="1Body">Selection of the coordinate system is essential for rotation problems. Otherwise, mistakes may occur due to inaccurate measurement of angular speed. Approximate inertial coordinate system selections for rotation problems should be the gravitational field of the celestial body higher than the object being rotated: (1) the Earth fixed Cartesian coordinate system for normal rotation problem; (2) heliocentric - geocentric Cartesian coordinate system for satellites orbiting the Earth; (3) the Galaxy Heart - heliocentric Cartesian coordinates for Earth's rotation around the Sun. In astrophysics, mass calculation error and angular velocity measurement error lead to a black hole conjecture.</p>

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