Closer look at the response of driven oscillators

In lecture we derived an expression for the complex amplitude $\underline{A} = Ae^{i\phi_o}$ of a driven and damped harmonic oscillator, with real amplitude $A$ and initial phase $\phi_o$,

$$A = \left\vert\underline{A}\right\vert = \frac{\frac{F_o}{... ...an^{-1} \left(\frac{b\omega}{\omega^2-\omega^2_o}\right)~ \;,$$ (1)

where $\omega_o \equiv \sqrt{k/m}$ is the natural frequency of the system.

In order to better appreciate the physical significance of these quantities and their frequency dependence, it is useful to plot them versus frequency and to identify certain important parameters of these plots. In this problem you will do this, starting from analyzing $A(\omega)/\frac{F_o}{m}$ and $\phi_o(\omega)$ in various limits¹:

(a)

Evaluate $A(\omega=0)/\frac{F_o}{m}$ and $\phi_o(\omega=0)$.

(b)

Evaluate $\lim_{\omega\rightarrow\infty} A(\omega)/\frac{F_o}{m}$ and $\lim_{\omega\rightarrow\infty} \phi_o(\omega)$.

(c)

Show that $A(\omega)/\frac{F_o}{m}$ has a maximum at $\omega=\omega_R=\sqrt{\omega_o^2-b^2/2}$ if $b^2 < 2\omega_o^2$. Express the value of ${(A/\frac{F_o}{m})}$ at the maximum in terms of $\omega_o$ and $b$.

Hint: Argue that the ratio of two positive expressions has a maximum when the denominator has a minimum. Then show that the denominator of $A$ in eq. (1) has a minimum at $\omega=\omega_R$ by checking that ${\left.\frac{dD(\omega)}{d\omega}\right\vert}_{\omega=\omega_R}\hspace{-3mm}=0$ for the function $D(\omega)\equiv(\omega^2_o-\omega^2)^2 + (b\omega)^2$.

(d)

In the limit $b \ll \omega_o$, show that $\omega_R \approx \omega_o$ and that $A(\omega)/\frac{F_o}{m}$ is approximately equal to ${(A/\frac{F_o}{m})}_{\rm max}/\sqrt{2}$ at each of the frequencies $\omega_{-} = \omega_o - b/2$ and $\omega_{+}=\omega_o+b/2$. Compute the `frequency band-width' $\Delta\omega \equiv \omega_{+}-\omega_{-}$.

(e)

Sketch plots of the functions $A(\omega)/\frac{F_o}{m}$ and $\phi_o(\omega)$ in the limit $b \ll \omega_o$. Label the quantities $\omega_o$, $(A/\frac{F_o}{m})_{\rm max}$, and $\Delta\omega$ on your plot.
Note: For this and other plots, you may plot the function on a plotting calculator or computer and then just sketch the basic shape of what you see. (You need not obtain a printout.) However, it is important that you label key points on your sketches (such as the position, width and height of the maxima) in terms of the basic quantities in the problem.