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sulla quale posizioniamo una massa vincolata ad un punto fisso tramite una molla ideale. \end{defJ} -NOTA: La molla è definita ideale in quanto non ha massa e rispetta perfettamente la legge di Hooke. +NOTA: La molla è definita ideale in quanto non ha massa e rispetta la legge di Hooke. Definito questo modello di base iniziamone lo studio. Immaginiamo di spostare la massa dalla posizione di equlibrio della molla. @@ -202,13 +202,13 @@ Esiste anche un'altra grandezza ricavata da $\omega_p$, che è strettamente lega \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 chiare in un espansione di Taylor-McLawrin: +Per condizioni di stabilità sappiamo che il potenziale $U$ ha delle caratteristiche ben precise in un espansione di Taylor-McLawrin: \begin{align*} - U(x) &= U(0) + \der{U(0)}{x} x + \frac 1 2 \der{U(0)}{x^2} x^2 + \dots\\ + U(x) &= U(0) + \derx{U(0)} x + \frac 1 2 \derxx{U(0)} x^2 + \dots\\ U(0) &= const.\\ - \der{U(0)}{x} &= 0\\ - \der{U(0)}{x^2} &> 0 + \derx{U(0)} &= 0\\ + \derxx{U(0)} &> 0 \end{align*} Ciò è intuitivo se pensiamo che per avere un oscillatore bisogna essere in una buca di potenziale. @@ -236,7 +236,7 @@ Studiamo l'energia meccanica totale e chiamiamo il paramentro $\alpha$ per parla \begin{align*} E_m = K + U\\ - K = \frac 1 2 m \dert{x}\\ + K = \frac 1 2 m \8\dert{x} \9^2\\ U = \frac 1 2 \alpha x^2 \end{align*} @@ -625,7 +625,257 @@ Scrivamo la formula in modo chiaro: -%%\section{Oscillatore armonico forzato} +\section{Oscillatore armonico forzato} +L'oscillatore smorzato è destinato a fermarsi, immaginiamo di dare energia al sistema. +Introduciamo quindi una forzate: +\[ + \dertt{x} + 2 \gamma \dert{x} + \omega_p^2 x = \frac{F_0}{m} \cos \omega_F t +\] + +Oscillatore forzato con forzante periodica o armonica. + +Anche in questo caso bisogna identificare una soluzione e la teoria delle equazioni differenziali ci dice che si ottiene sommando la soluzione generale per l'equazione differenziale omogenea associata e una soluzione particolare dell'equazione completa. + +Nel nostro caso noi già abbiamo la soluzione generale dell'omogenea associata: +\[ + x_0(t) = A e^{-\gamma t} \cos (\omega t + \varphi_0) +\] + +Perciò la soluzione generale sarà: +\[ + x(t) = A e^{-\gamma t} \cos (\omega t + \varphi_0) + x_p(t) +\] + +Come troviamo $x_p(t)$ ? Sicuramente ha una forma ben specifica: +\[ + x_p(t) = B \cos(\omega_f t + \varphi) +\] + +Quali valori di $B$ e $\varphi$, se esistono, devo sostituire? +\[ + x(t) = A e^{-\gamma t} \cos (\omega t + \varphi_0) + B \cos(\omega_f t + \varphi) +\] + +Questa è molto interessante perché per $ t >> \tau = \frac 1 \gamma$, superata la fase transiente, $x(t) = x_p(t)$. +In altre parole quando il sistema va a regime la componente che decade come un esponenziale è 0. +Tutto quello che resta è soltanto l'onda data dalla forzante. + +\begin{dimJ}{Dimostrazione della soluzione, parte 1}{dim:sol} + Introduciamo una funzione complessa \(z(t)\) tale che \(x(t)=\Re[z(t)]\). + Poiché \(\cos\alpha=\Re(e^{i\alpha})\), l’equazione reale equivale alla parte reale di + \[ + \dertt{z} + 2\gamma \dert{z} + \omega_p^2 z = \frac{F_0}{m} e^{i\omega_f t}. + \] + + Consideriamo prima l'equazione omogenea: + \[ + \dertt{z_h} + 2\gamma \dert{z_h} + \omega_p^2 z_h = 0. + \] + + La soluzione caratteristica è \(z_h(t)=C e^{(-\gamma+i\omega)t}\), dove \(\omega=\sqrt{\omega_p^2-\gamma^2}\). + Scrivendo \(C=A e^{i\varphi_0}\), otteniamo + \[ + z_h(t)=A e^{-\gamma t} e^{i(\omega t+\varphi_0)}. + \] + + Prendendo la parte reale: + \[ + x_h(t)=A e^{-\gamma t}\cos(\omega t+\varphi_0), + \] + che è il primo termine richiesto. +\end{dimJ} + +\begin{dimJ}{Dimostrazione della soluzione, parte 2}{dim:sol2} + Cerchiamo ora una soluzione particolare nella forma \(z_p(t)=K e^{i\omega_f t}\) + Calcoliamo le derivate: + \[ + \dert{z_p} = i\omega_f K e^{i\omega_f t}, \qquad \dertt{z_p} = -\omega_f^2 K e^{i\omega_f t} + \] + + Sostituendo nell’equazione complessa: + \[ + (-\omega_f^2 K + 2 i\gamma\omega_f K + \omega_p^2 K)e^{i\omega_f t} = \frac{F_0}{m} e^{i\omega_f t} + \] + + Dividendo per \(e^{i\omega_f t}\): + \[ + K(\omega_p^2 - \omega_f^2 + 2i\gamma\omega_f) = \frac{F_0}{m} + \] + + Da cui: + \[ + K = \frac{F_0/m}{\omega_p^2-\omega_f^2+2i\gamma\omega_f} + \] + + Scriviamo \(K = B e^{i\varphi}\). Allora: + \[ + z_p(t)=B e^{i(\omega_f t+\varphi)} + \] + + Prendendone la parte reale: + \[ + x_p(t)=B\cos(\omega_f t+\varphi) + \] + che coincide con il secondo termine della soluzione proposta. + + La soluzione complessiva è: + \[ + z(t)=A e^{(-\gamma+i\omega)t} e^{i\varphi_0} + B e^{i(\omega_f t+\varphi)} + \] + + Prendendo la parte reale si ottiene finalmente: + \[ + x(t)=A e^{-\gamma t}\cos(\omega t+\varphi_0) + B\cos(\omega_f t+\varphi) + \] + che soddisfa l’equazione originale. CVD. +\end{dimJ} + + + +\subsection{Condizioni Iniziali} +Sfruttiamo i risultati della dimostrazione. +Poniamo ora \(D = \omega_p^2-\omega_f^2 + 2 i \gamma \omega_f\) + +Il modulo del denominatore è +\[ + \abs{D} = \sqrt{(\omega_p^2 - \omega_f^2)^2 + (2\gamma\omega_f)^2} +\] + +Quindi il modulo di \(K\) è +\[ + B = \abs{K} = \frac{F_0/m}{\sqrt{(\omega_p^2 - \omega_f^2)^2 + (2\gamma\omega_f)^2}} +\] + +Scriviamo ora \(K = B e^{i\varphi}\), proprio come nella dimostrazione. +Per determinare l’angolo \(\varphi\) osserviamo che \(K\) è il reciproco (a meno di un fattore reale positivo \(F_0/m\)) del numero complesso +\[ + D = a + ib \qquad\text{con}\quad a = \omega_p^2 - \omega_f^2,\quad b = 2\gamma\omega_f +\] + +Il motivo per cui compare il reciproco risulta dalla soluzione particolare: si impone la forma \(z_p(t)=K e^{i\omega_f t}\), la si inserisce nell’equazione differenziale e i termini derivati producono un fattore complesso moltiplicativo che è proprio \(D\). +Per soddisfare l’equazione, questo fattore deve essere compensato dividendo per esso, dunque \(K\) è proporzionale a \(1/D\). +In altre parole, il numero complesso \(D\) “pesa” la risposta del sistema, e \(K\) deve contenerne il reciproco per annullarlo. + +Il numero \(D = a + ib\) può essere rappresentato come vettore nel piano complesso con una certa lunghezza \(\abs{D}\) e un certo angolo \(\theta\). +Questo angolo soddisfa: +\[ + \tan\theta = \frac{b}{a} +\] +perché il rapporto tra parte immaginaria e parte reale è la tangente dell’angolo del vettore. + +Dunque possiamo scrivere +\[ + D = \abs{D}\, e^{i\theta} +\] + +Il reciproco ha la forma +\[ + \frac{1}{D} = \frac{1}{\abs{D}} e^{-i\theta} +\] + +Qui si vede immediatamente che il reciproco inverte l’angolo del numero complesso. +Otteniamo quindi +\[ + \varphi = -\theta = -\arctan\!\left(\frac{2\gamma\omega_f}{\omega_p^2-\omega_f^2}\right) +\] + + + +\subsection{Casi particolari} +Compresa la natura delle condizioni iniziali possiamo fare una minima discussione di quello che succede per semplici limiti notevoli. + +\subsubsection*{Caso \(\gamma \sim 0\)} +Quando lo smorzamento è molto piccolo, il termine transitorio decade molto lentamente e il sistema si comporta come un oscillatore quasi conservativo. +In particolare, nel coefficiente di risposta stazionaria +\[ + B = \frac{F_0/m}{\sqrt{(\omega_p^2 - \omega_f^2)^2 + (2\gamma\omega_f)^2}} +\] + +il termine \(2\gamma\omega_f\) al denominatore diventa trascurabile tranne che in prossimità della risonanza. +Ne segue che, lontano dalla risonanza, la risposta del sistema è molto piccola, mentre in prossimità di \(\omega_f \sim \omega_p\) il denominatore tende a valori molto bassi e l’ampiezza cresce sensibilmente. +Nel limite \(\gamma \to 0\) la risonanza tende a diventare infinitamente “stretta” e il sistema oscilla con ampiezza molto grande quando la frequenza di forzamento coincide con quella naturale. + +\subsubsection*{Caso \(\omega_f \sim \omega_p\)} +Quando la frequenza di forzamento si avvicina alla frequenza naturale, il termine +\[ + (\omega_p^2 - \omega_f^2)^2 +\] +nel denominatore della risposta si riduce drasticamente. Il valore di \(B\) diventa quindi dominato dal termine di smorzamento +\[ + (2\gamma\omega_f)^2 +\] +e ciò porta a un picco di ampiezza attorno alla risonanza. +Per \(\gamma\) piccolo, questo picco è particolarmente accentuato e la curva della risposta assume la tipica forma risonante molto appuntita. +In sintesi, per \(\omega_f \sim \omega_p\) la risposta del sistema è grande e dominata quasi completamente dal regime forzato, mentre il transitorio diventa trascurabile dopo tempi brevi. + +\subsubsection*{Comportamento di \(\varphi\) per \(\omega_f \sim \omega_p\)} +La fase si ottiene da +\[ + \varphi = - \arctan\!\left(\frac{2\gamma\omega_f}{\omega_p^2 - \omega_f^2}\right) +\] + +Quando \(\omega_f\) si avvicina a \(\omega_p\), il termine \(\omega_p^2 - \omega_f^2\) tende a zero e il rapporto nella tangente diventa molto grande. +Ciò implica che l’angolo tende a +\[ + \varphi \to -\frac{\pi}{2} +\] +poiché la tangente cresce senza limite e l’arctangente tende a \(\pm \pi/2\) a seconda del segno. +Nel caso tipico con \(\gamma > 0\) e \(\omega_f\) crescente verso \(\omega_p\), l’argomento è positivo e dunque la fase risulta negativa e vicina a \(-\pi/2\). + +Fisicamente questo significa che, in prossimità della risonanza, l’oscillazione imposta dal forzamento è in ritardo di un quarto di periodo rispetto alla forza esterna, comportamento caratteristico degli oscillatori debolmente smorzati. +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: +\[ + x(t) = B \cos(\omega t + \varphi) +\] + +Come sempre sappiamo: +\[ + E_m = K + U \qquad + K = \frac 1 2 m \8\dert{x} \9^2 \quad + U = \frac 1 2 \alpha x^2 +\] + +Facciamo la derivata della soluzione: +\[ + \dert{x} = - B \omega \sin(\omega t + \varphi) +\] + +Sostituendo otteniamo: +\[ + K = \frac 1 2 m B^2 \omega^2 \sin^2(\omega t + \varphi) \quad + U = \frac 1 2 \alpha B^2 \cos^2(\omega t + \varphi) +\] + +Ricordiamo che \(\omega_p^2 = \frac \alpha m \imp \alpha = \omega_p^2 m\). +La soluzione risulta quindi: +\[ + E_m = K + U = \frac 1 2 m B^2 \8 \omega^2 \sin^2(\omega t + \varphi) + \omega_p^2 \cos^2(\omega t + \varphi) \9 +\] + +Questa formula sembra del tutto simile alla soluzione che avevamo trovato per il moto armonico semplice, ma i 2 termini non si compensano esattamente. +Ciò è ragionevole in quanto l'energia cinetica è data dal moto forzato, mentre l'energia elastica segue il moto della molla. +Infatti se la forzante fosse in risonanza(\(\omega = \omega_p\)): +\[ + E_m = \frac 1 2 m \omega_p^2 B^2 +\] + +Ritroviamo un oscillatore armonico semplice. +In questo modo stiamo dando l'energia istantaneamente quando viene persa con la forzante. + +Torniamo al caso generale: +\begin{align*} + E_m = K + U = \frac 1 2 m B^2 \8 \omega^2 \sin^2(\omega t + \varphi) + \omega_p^2 \cos^2(\omega t + \varphi) \9\\ + E_m = \frac 1 2 m B^2 \{ \omega^2 [1 - \cos^2(\omega t + \varphi)] + \omega_p^2 \cos^2(\omega t + \varphi) \}\\ + E_m = \frac 1 2 m B^2 \{ \omega^2 + (\omega_p^2 - \omega^2) \cos^2(\omega t + \varphi) \} +\end{align*} + +L'equazione dell'energia non decade ma oscilla tra 2 valori d'energia. +L'oscillatore è quindi un moto perpetuo, ma questo non va contro le leggi della fisica poiché al sistema viene continuamente data energia.