Dissipation of Energy and Q Factor

Subsection

13.7.1

Energy of an Underdamped Oscillator

Just as for any other system, the energy of a damped oscillator is the sum of its kinetic and potential energies. We will consider only the underdamped case, since, it is the only one that actually oscillates – the other two cases, the overdamped and the critically damped, do not oscillate.

\begin{equation}
E = \dfrac{1}{2}m v^2 + \dfrac{1}{2}k x^2,\label{eq-Energy-of-an-Underdamped-Oscillator-main}\tag{13.7.1}
\end{equation}

where \(x \) and \(v\) are for underdamped oscillator are

\begin{align*}
\amp x = A\,e^{-\gamma t/2}\, \cos( \omega^{\prime} t + \phi ), \\
\amp v = -\dfrac{\gamma}{2}\, x – \omega^{\prime}\,A\,e^{-\gamma t/2}\, \sin( \omega^{\prime} t + \phi ),
\end{align*}

When you insert these in Eq. (13.7.1), you get a really ugly expression, which shows that energy of the oscillator oscillates in time with decreasing amplitudes, but at twice the frequency of the oscillation of \(x \text{.}\)

But if you have a lightly damped oscillator, i.e., if \(\gamma \lt\lt \omega \text{,}\) meaning that the tendency to damp is considerably small compared to the tendency to oscillate, then, we can show that, on average, energy dies out exponentially with time.

\begin{equation}
E(t) \approx E(0)\, e^{-\gamma\, t},\label{eq-dissipation-of-energy-energy-t}\tag{13.7.2}
\end{equation}

where \(E(0) \) is the energy at initial time, which for a lightly damped oscillator is

\begin{equation*}
E(0) \approx \dfrac{1}{2}k A^2.
\end{equation*}

Eq. (13.7.2) shows that energy of a lightly damped oscillator reduces to about one-third in time, \(\tau\text{,}\) equal to inverse of \(\gamma\text{.}\)

\begin{equation*}
\tau = \dfrac{1}{\gamma}.
\end{equation*}

This \(\tau\) is sometimes called time constant for energy decay. It gives us time for energy to decay to \(1/e\approx 1/2.7\) of the original energy.