Energy in a Damped Harmonic Oscillator

Alexei Gilchrist

Energy of the damped harmonic oscillator is described. The Q of an oscillator.

1 Introduction

In another node (damped-harmonic-oscillator) we derived the motion of an under-damped harmonic oscillator and found

\[\begin{equation*}x(t) = A e^{-\gamma/2 t} \cos(\omega_d t+\phi),\end{equation*}\]
where \(\omega_d = \sqrt{\omega_0^2-\gamma^2/4}\), \(\gamma\) is the damping rate, and \(\omega_0\) is the angular frequency of the oscillator without damping. Here we will investigate the energy of the system. Clearly this is no longer a closed system so we should expect the energy to dissipate to the environment and the motion to cease eventually.

2 Energy in the underdamped oscillator

Differentiating the position we get the velocity

\[\begin{equation*}\dot{x}(t) = -\frac{1}{2} A e^{-\gamma/2 t}\left[ \gamma\cos(\omega_d t+\phi) + 2\omega_d\sin(\omega_d t+\phi) \right].\end{equation*}\]

Looking at the total mechanical energy (sum of the kinetic and potential energy terms), we’d expect this decay away with time as the velocity dependent damping is removing energy from the mechanical system. The kinetic energy is given as usual by \(E_k=\frac{1}{2}m\dot{x}^2\) and the potential by \(E_p=\frac{1}{2}kx^2 = \frac{1}{2}m \omega_0^2 x^2\), where we have used \(\omega_0^2=k/m\). Using the position and velocity equations we derived yields

\[\begin{align*}E_p &= \frac{1}{2} m A^2 \omega_0^2 e^{-\gamma t}\cos^2(\omega_d t+\phi), \\ E_k &= \frac{1}{8} m A^2 e^{-\gamma t} \left[ \gamma\cos(\omega_d t+\phi) + 2\omega_d\sin(\omega_d t+\phi) \right]^2.\end{align*}\]
hence the total energy is
\[\begin{align*}E &= E_k+E_p = \frac{1}{8} m A^2 e^{-\gamma t}\left(\left[ \gamma\cos(\omega_d t+\phi) + 2\omega_d\sin(\omega_d t+\phi) \right]^2+ 4 \omega_0^2\cos^2(\omega_d t+\phi)\right) \\ & = \frac{1}{8} m A^2 e^{-\gamma t} \left( 4\omega_0^2 + 2\gamma \omega_d \sin(2\omega_d t+2\phi) + \gamma^2\cos(2\omega_d t+2\phi)\right) \\ & = \frac{1}{2} m A^2\omega_0^2 e^{-\gamma t}\left[1+\frac{\gamma}{2\omega_0}\cos(2\omega_d t+\phi')\right],\end{align*}\]
and \(\phi' = -\tan^{-1}(2\omega_d/\gamma)\).

You can see that it is modulated by an overall decay \(e^{-\gamma t}\). For \(\gamma\ll\omega_0\) (weak damping limit) we can neglect the cosine term to get

\[\begin{equation*}E \approx \frac{1}{2} m A^2\omega_0^2 e^{-\gamma t},\end{equation*}\]
and in the limit of no damping (\(\gamma\rightarrow 0\)) the total energy becomes the constant \(1/2 m A^2\omega_0^2\) as expected (see harmonic-oscillator).

In general however, the damping depends on the velocity and since the velocity is changing with time we should expect the loss of energy from the system to also show oscillations. The cosine term oscillates with the angular frequency \(2\omega_d\) which is twice the frequency of the oscillator. Take a moment to think why this is the case.

The energy of a damped harmonic oscillator.

The behaviour of the energy is clearly seen in the graph above. The period of oscillation is marked by vertical lines. You can see that the rate of loss of energy is greatest at 1/4 and 3/4 of a period. This corresponds to the times of largest velocity and hence largest damping. The overall trend is the exponential decay and as \(\gamma\) becomes small the oscillations become less discernible on a timescale for an equivalent decay:

The energy of a weakly damped harmonic oscillator.

3 Q-factor

The quality factor (\(Q\) factor) is a dimensionless parameter quantifying how good an oscillator is. The less damping the higher the \(Q\) factor. It is defined as the number of radians that the oscillator undergoes as the energy of the oscillator drops from some initial value \(E_0\) to a value \(E_0e^{-1}\).

For low damping (small \(\gamma/\omega_0\)) the energy of the oscillator is approximately

\[\begin{equation*}E(t) = E_0 e^{-\gamma t},\end{equation*}\]
hence \(E(t)=E_0 e^{-1}\) when \(t=1/\gamma\). Over this time the oscillator will undergo \(t/T\) cycles (where \(T\) is the period) and each is \(2\pi\) radians. Since \(T=2\pi/\omega_d\) we have
\[\begin{equation*}Q = \frac{\omega_d}{\gamma} \approx \frac{\omega_0}{\gamma}.\end{equation*}\]
The final approximation is justified because we are considering the weak damping regime and \(\omega_d=\sqrt{\omega_0^2-\gamma^2/4}\approx \omega_0\).

Alternatively we can imagine looking at the oscillator’s energy only at a particular point in its cycle, and since the cosine term will always evaluate to the same value at those points, the total energy will fall as

\[\begin{equation*}E(t) = E_0 e^{-\gamma t}\end{equation*}\]
with some constant \(E_0\). Examining the energy lost to the environment in one cycle we have
\[\begin{equation*}\Delta E(t) = E(t) - E(t+T) = E(t) - E(t+2\pi/\omega_d) = E_0 e^{-\gamma t}\left(1-e^{-\frac{2\pi\gamma}{\omega_d}}\right) = E_0 e^{-\gamma t}\left(1-e^{-\frac{2\pi}{Q}}\right)\end{equation*}\]
hence the fraction of energy lost per cycle is
\[\begin{equation*}\frac{\Delta E(t)}{E(t)} = 1-e^{-\frac{2\pi}{Q}}.\end{equation*}\]
Note that this is constant.

A “high-Q” oscillator is a good oscillator—one that has very low damping, a “low-Q” oscillator is one that rapidly leaks its energy to the environment.

© Copyright 2021 Alexei Gilchrist