Complex numbers and ODEs: problem set 3 (differential equations for oscillators)

2. Damped oscillators without forcing

A damped harmonic oscillator is displaced by a distance $x_0$ and released at a time $t = 0$. Show that subsequent motion is described by the differential equation \[ m \frac{\de^2 x}{\de t^2} + m\gamma \frac{\de x}{\de t} + m\omega_0^2 x = 0. \]

with $x(0) = x_0$ and $x'(0) = 0$.

a) Damped solutions

Find and sketch solutions for (i) overdamping ,(ii) critical damping, and (iii) underdamping.


b) Deriving the Q-Factor

For a lightly damped oscillator, the quality factor, or Q-factor, is defined as \[ Q = \frac{\text{energy stored}}{\text{energy lost per radian of oscillation}}. \]

Show that $Q = \omega_0/\gamma$.


5. Damped oscillators with forcing

Consider the damped oscillator subject to an oscillatory driving force: \[ m \frac{\de^2 x}{\de t^2} + m\gamma \frac{\de x}{\de t} + m\omega_0^2 x = F\cos \omega t. \]

a) Calculate the steady state solution

Explain what is meant by the steady state solution of this equation and calculate the steady state solution for the displacement, $x(t)$ and the velocity, $v(t) = x'(t)$.


b) The displacement and velocity

Sketch the amplitudes and phases of the displacement and velocity as a function of $\omega$.


c) Resonant frequencies

Determine the resonant frequency for both the displacement and the velocity.


d) Full width at half maximum of resonance (in the amplitude)

Defining $\Delta \omega$ as the full width at half maximum of the resonance peak, calculate $\Delta\omega/\omega_0$ to leading order in $\gamma/\omega_0$


e) The power of the system

For a lightly damped, driven oscillator near resonance, calculate the energy stored and the power supplied to the system. Hence confirm that $Q = \omega_0/\gamma$. How is $Q$ related to the width of the resonance peak?