El problema de Riemann para una cuerda elástica con ley de Hooke lineal
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Date
2017
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Universidad Nacional de Trujillo
Abstract
En este trabajo, hemos investigado el problema del movimiento horizontal y vertical
de una cuerda infinita, elástica y homogénea con una relación entre la tensión y
la deformación; que es modelada, usando la segunda ley de Newton, por un sistema
2 2 de ecuaciones diferenciales parciales de segundo orden. Para la formulación del
problema de Riemann, se transformó el sistema anterior en un sistema 4 4 de leyes de
conservación y se consideró una condición inicial discontinua formada por dos estados
constantes. Se consideró además, una relación lineal (ley de Hooke) entre la tensión y
la deformación de la cuerda. Para resolver este problema, se calcularon los autovalores
y autovectores de la matriz asociada al sistema de leyes de conservación, las ondas de
rarefacción y ondas de choque que determinan los estados constantes de la solución
del problema planteado.
In this work, we have investigated the problem of horizontal and vertical movement of the infinite, elastic and homogeneous string with a relation between tension and deformation; which is modeled, using Newton’s second law, by a 2 2 system of second-order partial differential equations. For the formulation of the Riemann problem, the previous system was transformed into a 4 4 system of conservation laws and was considered a discontinuous initial condition formed by two constant states. A linear relationship (Hooke’s law) was also considered between tension and deformation of the string. To solve this problem, we calculated the eigenvalues and eigenvectors of the matrix associated with the conservation law system, rarefaction waves and shock waves that determine the constant states of the solution of the raised problem.
In this work, we have investigated the problem of horizontal and vertical movement of the infinite, elastic and homogeneous string with a relation between tension and deformation; which is modeled, using Newton’s second law, by a 2 2 system of second-order partial differential equations. For the formulation of the Riemann problem, the previous system was transformed into a 4 4 system of conservation laws and was considered a discontinuous initial condition formed by two constant states. A linear relationship (Hooke’s law) was also considered between tension and deformation of the string. To solve this problem, we calculated the eigenvalues and eigenvectors of the matrix associated with the conservation law system, rarefaction waves and shock waves that determine the constant states of the solution of the raised problem.
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Keywords
Problema de Riemann, Cuerda elástica, Leyes de conservación, Ondas de choque, Ondas de rarefacción