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zr}R?ZzHbDF#>1H*`_T)LbUW%WWqUFeEhB6j1N9X7nRM+7q zty+_Cue>ckrL^TL#p)1i_$^4YaX==wA|;KxGtKspL6LAvjtYoGJy7rVOEVaa->~$8 z9Ooz3DsBkZG;W+WH#+Fl_!U^jXs!6br1(E!>f~Sc|A|x1l>2Ij6D+#MpU_mOv&jE9 ip*fiSCylwf8N0Z7xqM|XR#t8<4i-2H3JE1ixc>tCF^?|* diff --git a/onde/onde.tex b/onde/onde.tex index c91d51d..1461cb1 100644 --- a/onde/onde.tex +++ b/onde/onde.tex @@ -200,6 +200,7 @@ Esiste anche un'altra grandezza ricavata da $\omega_p$, che è strettamente lega \end{defJ} + \section{Energia Oscillatore Armonico} Facciamo lo studio dell'energia di un oscillatore armonico. Per condizioni di stabilità sappiamo che il potenziale $U$ ha delle caratteristiche ben precise in un espansione di Taylor-McLawrin: @@ -625,6 +626,7 @@ Scrivamo la formula in modo chiaro: + \section{Oscillatore armonico forzato} L'oscillatore smorzato è destinato a fermarsi, immaginiamo di dare energia al sistema. Introduciamo quindi una forzate: @@ -827,6 +829,7 @@ Si dice che oscillazione e forzamento sono in quadratura di fase. + \section{Energia oscillatore armonico forzato} Per situazioni \(\tau >> \frac 1 \gamma\) vale la soluzione: \[ @@ -879,4 +882,132 @@ L'oscillatore è quindi un moto perpetuo, ma questo non va contro le leggi della +\section{Sistemi a più gradi di libertà} +Partiamo da un modello molto semplice: 2 pendoli legati da una molla. +I 2 pendoli sarebbero totalmente indipendenti se non fosse per la molla. +Facciamo un po' di nomencatura: +\begin{itemize} + \item $m$: massa delle molle + \item $K$: costante elastica della molla + \item $x_1$: spostamento della massa 1 dall'equilibrio + \item $x_2$: spostamento della massa 2 dall'equilibrio + \item $l$: lunghezza del pendolo +\end{itemize} + +Possiamo scrivere: +\[ + F_1 = - \frac{mg}{l} x_1 - K(x_1 - x_2) \qquad + F_2 = - \frac{mg}{l} x_2 - K(x_2 - x_1) +\] + +Scriviamo le equazioni del moto per le 2 masse: +\[ + m \dertt{x_1} = - \frac{mg}{l} x_1 - K(x_1 - x_2) \qquad + m \dertt{x_2} = - \frac{mg}{l} x_2 - K(x_2 - x_1) +\] + +Sommiamo e sotraiamo le 2 equazioni: +\begin{align*} + m \8 \dertt{x_1} + \dertt{x_2} \9 = - \frac{mg}{l} (x_1 + x_2)\\ + m \8 \dertt{x_1} - \dertt{x_2} \9 = - \frac{mg}{l} (x_1 - x_2) - 2 K (x_1 - x_2) +\end{align*} + +A questo punto definisco $u_a = x_1 + x_2$ e $u_b = x_1 - x_2$. +Otteniamo 2 equazioni: +\begin{align*} + m \dertt{u_a} = - \frac{mg}{l} u_a\\ + m \dertt{u_b} = - \frac{mg}{l} u_b - 2 K u_b +\end{align*} + +Queste sono equazioni sono indipendenti e sappiamo come risolverle. +\begin{align*} + \dertt{u_a} + \frac g l u_a = 0 \qquad \dertt{u_a} + \omega_a^2 u_a = 0\\ + \dertt{u_b} + \8 \frac{g}{l} + 2 \frac K m \9 u_b = 0 \qquad \dertt{u_b} + \omega_b^2 u_b = 0 +\end{align*} + +Le soluzioni sono quindi: +\[ + u_a = A_a \cos(\omega_a t + \varphi_a) \qquad + u_b = A_b \cos(\omega_b t + \varphi_b) +\] + +Ora noi dobbiamo tornare indietro a $x_1$, $x_2$. +\begin{align*} + x_1(t) = \frac{u_a + u_b}{2} \\ + x_2(t) = \frac{u_a - u_b}{2} +\end{align*} + +Abbiamo completato lo studio del sistema. +Se osserviamo i moti dei pendoli questi sono la sovrapposizione di 2 moti semplici. +Nel modo $u_a$ le 2 masse si muovono insieme e la molla non fa nulla, mentre nel modo $u_b$ i pendoli si contraggono e si aprono in modo simmetrico. +Il nostro moto complesso diventa facile da studiare se pensato come la sovrapposizione di 2 moti disitinti e indipendenti. +Questi sono chiamati MODI NORMALI, nel caso specifico informalmente sono chiamati modo a pendolo e modo a respiro. + + + +\section{Battimenti} +Pendiamo l'esempio precedente e caliamoci nel caso specifico dove il pendolo parte da fermo con le seguenti posizioni iniziali: +\[ + x_1(0) = -2A \qquad + x_2(0) = 0 +\] + +Se noi vogliamo studiare quello che stà succdedendo dobbiamo scomporre la nostra condizione iniziale nei 2 modi semplici. +Facciamo una prova ragionata: prendendo il modo a pendolo e mettendolo a $-A$ avremmo che +\[ + x_1(0) = -A \qquad + x_2(0) = -A +\] + +A questo punto introduciamo un modo a respiro che sia "allargato" di una A. Il nostro modo a respiro avrebbe condizione iniziale: +\[ + x_1(0) = -A \qquad + x_2(0) = +A +\] + +La somma dei 2 modi all'istante iniziale da la condizione iniziale: +\[ + x_1(0) = -2A \qquad + x_2(0) = 0 +\] + +Abbiamo quindi scomposto il caso iniziale nei suoi 2 modi normali. +\[ + \omega_a = \sqrt{\frac g l} \qquad + \omega_b = \sqrt{\frac g l + 2 \frac k m} +\] + +Se osserviamo il moto della massa 1 $x_1$ ci accorigamo che è la somma di 2 oscillazioni simili ma non uguali. +L'effetto prodotto è quello di un oscillazione che ogni tanto interferisce in modo costruttivo ed ogni tanto in modo distruttivo. +Questo fenomeno si chiamano BATTIMENTI. +\[ + x_1(t) = C_a \cos(\omega_a t + \varphi_a) + C_b \cos(\omega_b t + \varphi_b) +\] + +Nel caso specifico: $\varphi_a = \varphi_b = 0$, $C_a=C_b$: +\[ + x_1(t) = A [\cos (\omega_a t) + \cos (\omega_b t)] +\] + +Usiamo le formule di Prostaferesi: +\[ + \cos\alpha + \cos\beta = 2 \cos\!\left(\frac{\alpha + \beta}{2}\right) \cos\!\left(\frac{\alpha - \beta}{2}\right) \qquad + \alpha = \omega_a t \quad \beta = \omega_b t +\] + +Definiamo 2 variabili con nomi che sembrano casuali, ma diventerranno chiari: +\[ + \frac{\alpha + \beta}{2} = \frac{\omega_a + \omega_b}{2} t = \omega_M t \qquad + \frac{\alpha - \beta}{2} = \frac{\omega_a - \omega_b}{2} t = \omega_{inv} t +\] + +Ne risulta che: +\[ + x_1(t) = 2 \cos(\omega_M t) \cos(\omega_{inv} t) +\] + +Risuta chiaro che $\omega_{inv} << \omega_M$. +Se noi grafichiamo questo andamento ci accorgeremmo che $\omega_{inv}$ rappresenta l'inviluppo della curva, +mentre $\omega_M$ le oscillazioni più piccole che avvengono nel "range" definito dall'inviluppo. + \end{document}