MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C61CDF.7A4980C0" Este documento es una página Web de un solo archivo, también conocido como archivo de almacenamiento Web. Si está viendo este mensaje, su explorador o editor no admite archivos de almacenamiento Web. Descargue un explorador que admita este tipo de archivos, como Microsoft Internet Explorer. ------=_NextPart_01C61CDF.7A4980C0 Content-Location: file:///C:/59165C93/MetododeDiferenciasFinitas.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Método de Diferencias Finitas

1 Método de Diferencias Finitas

El método de diferencias finitas es una clásica aproximación para encontrar la solución numérica de las ecuaciones que gobiernan el modelo matemático= de un sistema continuo. Es valioso familiarizarse con ésta aproximación porque tal conocimiento reforzará la comprensión de los procedimientos de elementos finitos.

Básicamente, en una solución por diferen= cias finitas, las derivadas son reemplazadas por aproximaciones en diferencias finitas, convirtien= do entonces un problema de ecuaciones diferenciales en un problema algebraico fácilmente resoluble por medios comunes (especialmente matriciales).=

1.1 Método de la Expansión de Taylor

El méto= do de expansión de Taylor es una forma alternativa de obtener aproximaciones de diferencia. Este método no solo deduce las fórmulas de diferencia sistemáticamente, sino que tambi&eacut= e;n deduce los términos de error.

Para una derivada de p-ésimo orden, el nú= ;mero mínimo de puntos de datos requeridos para deducir una aproximación de diferencia es , así por ejemplo una aproximación de diferencia para la primera derivada de una función necesita por lo m= enos de dos puntos de datos.

Consideremos la deducción de la aproximaci&oacu= te;n de diferencia para en términos de . La expansión de Taylor de alrededor d= e es .

Resolvie= ndo la ecuación anterior para la primera derivada, tenemos

Si ignoramos todos los términos con excepci&oac= ute;n del primero del miembro derecho de la ecuación anterior, obtendremos= la aproximación por diferencia = hacia adelante. Los términos que se ignoran constituyen el error de truncado, representado por el término inicial, .

La aproximación de diferencia hacia adelante, c= on el error de truncado incluido, se expresa como (3), d&oacu= te;nde .

1.2 Expresiones de diferencias, cuyo uso es frecuente.

Primera Deriva= da

Aproximaciones= de diferencia hacia adelante

Aproximaciones de diferencia hacia atrás

Segunda Deriva= da

Aproximaciones= de diferencias hacia adelante

Aproximaciones de diferencia hacia atrás

1.3&= nbsp; Simulación de la Propagación de Onda

1.3.1 Introducción a las oscilaciones

Una oscilación es un movimiento periódico en torno de un pun= to de equilibrio estable. L= as ecuaciones de movimiento del oscilador armónico simple son:

<= o:p>

<= sub> donde w es la frecuencia de angular= .

<= o:p>

c= on la solución general:

<= o:p>

<= sub>

<= o:p>

y= la solución específica

<= o:p>

<= sub> dond= e d es el&n= bsp; ángulo de desfase de la oscilación.<= /p>

<= o:p>

s= i la condición inicial es

<= o:p>

<= sub>

<= o:p>

El oscilador es amortiguado, cuando existe una fuerza de roce que le hace perder energía. La ecuación de movimiento se cambia a:

donde g es el coeficiente de amortiguamiento de la oscilación.

Se distinguen los casos sobre-amortiguado, amortiguamiento crítico y amortiguado, dependiendo de la relación de la fuerza de roce y el sistema del oscilador. Sólo en el último caso hay oscilaciones, con la amplitud decayendo de manera exponencial. La solución para este caso es:=

El oscilador forzado resulta, cuando una fuerza externa periódica con frecuencia w<= /span> actúa sobre un oscilador armónico. La ecuación de movimiento en este caso es:

con la solución

Cuando la frecuencia w<= /span> de la fuerza externa es igual a la frecuencia característica w₀ del sistema, se habla de resonancia. En este caso, la amplitud logra su valor máximo: (sólo para amortiguamiento débil)

Cuando la amplitud F0 de la fuerza externa es grande y el amortiguamiento es débil, puede resultar una resona= ncia catastrófica, donde la energía de la oscilación e= xcede la energía adhesiva del material con el resultado que és= te queda destruido. Figura 41.

Figura 41 Puente de Tacoma. Colapsó en 1940 por en= trar en resonancia con una ráfaga de viento.

1.3.2 Ecuación de Onda

Un pulso se puede describir por la funció= ;n , si el pulso no se deforma al propagarse (medio no dispersivo) se puede describir como:

Figura SEQ Figura \* ARABIC 42 Pulso viajero

La onda unidimensional (Figura 43), no dispersiva que viaja con velocidad n, se puede describir por:

Figura 43 Onda unidimensional

con el punt= o O como referencia.

con el punto O’ como referencia.

Con el sistema de coordenadas movi&e= acute;ndose junto con el pulso con velocidad n<= /span>.

en general:

, si la onda viaja hacía la derecha.

, si la onda viaja hacía la izquierda.

La forma general para una onda de una dimensión , es función de dos variables.

Usando derivadas parciales:

es la ecuación de onda en una dimensión.

1.3.3 Método de diferencias finitas para la ecuación de onda en una dimensión

La ecuación de ondas en una dimensión ap= arece en el análisis de una gran cantidad de fenómenos físic= os, desde las vibraciones de una cuerda hasta sistemas de osciladores acoplados, propagación de luz en medios, ... La forma de esta ecuación e= s:

donde representa la velocidad de propagación de la onda = en el punto y el instante . Esta ecuación debe complementarse con unas condiciones de contorno y/o iniciales apropiadas. Por ejemplo el problema p= uro de valores iniciales requiere especificar:

que puede suplementarse con unas condiciones de contorno del tipo:

El esquema de solución apropiado para la ecuación diferencial de la propagación de onda es:

Este esquema es completamente explícito pero re= quiere conocer la solución en dos niveles temporales para poder arrancar. <= /p>

El esquema de diferencias finitas así construido tiene un error de truncación desde el segundo término, ya que solo aproxima la ecuación con una serie de dos términos.

Existen requerimientos de estabilidad para el esquema de diferencias finitas. Uno de ellos es , donde Dt es el paso en el tiempo y Dx es el tamaño del paso en el espacio y v es = la velocidad de propagación.

Con el fin de disminuir la dispersión numérica es necesario lograr que la longitud de on= da de la señal esté contenida en no menos de 5 pasos en el espac= io y no más de 11. Por lo q= ue el periodo calculado como el inverso de la frecuencia , multiplicado por la velocidad de propagación de = la onda debe ser igual a la longitud de 5 a 11 pasos en el espacio.

Es por esto que los parámetros para la solución de la ecuación de onda, como el paso en el tiempo y = el espacio, por medio del esquema de diferencias finitas, dependen de la frecuencia de la fuente que transmite la señal y de la velocidad de propagación del medio en el que es transmitida.

Algunos rangos de frecuencia y de velocidad de propagación= son mostrados en la Tabla 1 y en la Tabla 2 .

Fuente	Frecuencia
Trueno	20 Hz a 8= 0 Hz
Hablar, e= l sonido de la voz	250 Hz a = 6,000 Hz
Una canta= nte de opera en un tono alto	2,000 Hz

Tabla 1 Frecuencia de algunos sonidos comunes

Medio	Velocidad de Propagaci&= oacute;n del sonido
Aire	344 m/s
Agua	1440m/s
Acero	= 5000 m/s

Tabla 2 Velocidad de propagació= n en diferentes medios

Problema 7=

Usar el código en Matlab de la función o= nda1D, que implementa el esquema de diferencias finitas, ubicado en la dirección:

http://laboratoriomatematicas.uniandes.edu.co/semarqui= tec/onda1D.m

Para simular la propagación del sonido generado= por un trueno y la voz de una persona, a través del aire y del agua.

1.3.4 Principio de Huygens

El principio de Huygens afirma que todo punto de un fr= ente de onda inicial puede considerarse como una fuente de ondas esféricas secundarias que se extienden en todas las direcciones con la misma velocida= d, frecuencia y longitud de onda que el frente de onda del que proceden. Con e= llo puede definirse un nuevo frente de onda que envuelve las ondas secundarias. Cando las ondas secundarias llegan a otro medio u objeto, cada punto del límite entre los medios se convierte en una fuente de dos conjuntos de ondas. El conjunto reflejado vuelve al primer medio, y el conjunto refractado entra en el segu= ndo medio. Por esta razón, el diseño de los espacios con fines acústicos, como los teatros o salas de presentación, contempla los efectos de las superficies en la generación de nuevas ondas acústicas.

Figura SEQ Figura \* ARABIC 44 Generación de nuevas fuentes de onda

Ejercicio

Comprobar los efectos de los obstáculos en los frentes de onda y simular la superposición de ondas.

Para esto es necesario descargar el programa ubicado e= n la dirección:

http://laboratoriomatematicas.uniandes.edu= .co/semarquitec/waves.zip

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