Usuario:Javier

Javier Ortiz Torres Fenomenos Ondulatorios javier19df@hotmail.com --Javier 20:59 8 feb 2009 (CST)

Oscilaciones Forzadas

Sea A un istema de masa m sujeto a un resorte ideal que obedece la ley de Hooke. Si el sistema se encuentra inmerso en un medio resistente que ejerce una fuerza de amortiguamiento proporcinal a la primera potencia de la velocidad y si además se ejerce sobre él una fuerza de la forma $f(x)=f_{0}cos\left(w_{0}t\right)$ (véase Fig.1) entonces por segunda ley de Newton tenemos que:

$-\propto {\ddot {x}}-kx+f_{0}cos\left(w_{0}t\right)=m{\ddot {x}}$

o bien

$m{\ddot {x}}+\propto {\dot {x}}+kx=f_{0}cos\left(w_{0}t\right)$

Si definimos

$\beta \equiv {\frac {\propto }{2m}},w\equiv {\frac {k}{m}},F_{0}\equiv {\frac {f_{0}}{m}}$

obtenemos la siguiente ecuacion diferencial

${\ddot {x}}+2\beta {\ddot {x}}+wx=F_{0}cos\left(w_{0}t\right)$

Ahora bien fijamos sobre el sistema una cuerda ideal de longitud l de manera que $x=\psi$ obetenemos

${\ddot {\psi }}+2\beta {\dot {\psi }}+w_{0}\psi =F_{0}cos\left(w_{0}t\right)$

Cuya solucion es la suma de la ecuacion

$\psi _{general}=\psi _{homogenea}+\psi _{particular}$

${\ddot {\psi }}_{c}+2\beta {\dot {\psi }}_{c}+w_{0}\psi _{c}=F_{c}$

$\psi _{c}=\psi _{0}e^{iw_{0}t}$

${\dot {\psi _{c}}}=i\psi _{0}we^{iw_{0}t}$

${\ddot {\psi _{c}}}=-\psi _{0}w^{2}e^{iw_{0}t}$

$(w_{0}^{2}-w^{2}+2\beta iw)\psi _{0}e^{iw_{0}t}=F_{c}$

$(w_{0}^{2}-w^{2}+2\beta iw)\psi _{c}=F_{c}$

$\psi _{c}={\frac {F_{0}e^{iwt}}{w_{0}^{2}-w^{2}+2\beta iw}}$

$\psi _{c}\equiv {\frac {F_{0}}{w_{0}^{2}-w^{2}+2\beta iw}}$

$\psi _{c}={\frac {F_{0}}{w_{0}^{2}-w^{2}+2\beta iw}}{\frac {(w_{0}^{2}-w^{2}+2\beta iw)}{(w_{0}^{2}-w^{2}+2\beta iw)}}={\frac {F_{0}(w_{0}^{2}-w^{2}+2\beta iw)}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}$

$\psi _{c}={\frac {F_{0}(w_{0}^{2}-w^{2})}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}-i{\frac {F_{0}2\beta w}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}=\psi _{0}e^{i\propto }$

$\psi _{0}(w)={\frac {1}{\sqrt {(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}}$

$\propto (w)=ArcTan({\frac {-2\beta w}{w_{0}^{2}-w^{2}}})$

El estado estable

${\ddot {\psi }}_{c}+2\beta {\dot {\psi }}_{c}+w_{0}\psi _{c}=F_{c}$

$\psi _{c}=\psi _{0}e^{iw_{0}t}$

${\dot {\psi _{c}}}=i\psi _{0}we^{iw_{0}t}$

${\ddot {\psi _{c}}}=-\psi _{0}w^{2}e^{iw_{0}t}$

$(w_{0}^{2}-w^{2}+2\beta iw)\psi _{0}e^{iw_{0}t}=F_{c}$

$(w_{0}^{2}-w^{2}+2\beta iw)\psi _{c}=F_{c}$

$\psi _{c}={\frac {F_{0}e^{iwt}}{w_{0}^{2}-w^{2}+2\beta iw}}$

$\psi _{c}\equiv {\frac {F_{0}}{w_{0}^{2}-w^{2}+2\beta iw}}$

$\psi _{c}={\frac {F_{0}}{w_{0}^{2}-w^{2}+2\beta iw}}{\frac {(w_{0}^{2}-w^{2}+2\beta iw)}{(w_{0}^{2}-w^{2}+2\beta iw)}}={\frac {F_{0}(w_{0}^{2}-w^{2}+2\beta iw)}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}$

$\psi _{c}={\frac {F_{0}(w_{0}^{2}-w^{2})}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}-i{\frac {F_{0}2\beta w}{(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}=\psi _{0}e^{i\propto }$

$\psi _{0}(w)={\frac {1}{\sqrt {(w_{0}^{2}-w^{2})^{2}+4\beta ^{2}w^{2}}}}$

$\propto (w)=ArcTan({\frac {-2\beta w}{w_{0}^{2}-w^{2}}})$