22.3: Resonance in a Damped Driven Linear Oscillator- A Brief Review

This is just to remind you of what we covered in lecture 18, before we add anharmonic terms in the next section.

The linear damped driven oscillator has equation of motion:

\[\ddot{x}+2 \lambda \dot{x}+\omega_{0}^{2} x=(f / m) e^{i \gamma t}\]

(Following Landau’s notation here note it means the actual frictional drag force is \(2 \lambda m \dot{x}\))

Looking near resonance for steady state solutions at the driving frequency, with amplitude \(b\), phase lag \(\delta\), that is, \(x(t)=b e^{i(\gamma t+\delta)}\), we find

\[b e^{i \delta}\left(-\gamma^{2}+2 i \lambda \gamma+\omega_{0}^{2}\right)=(f / m)\]

For a near-resonant driving frequency

\[\gamma=\omega_{0}+\varepsilon\]

and assuming the damping to be sufficiently small that we can drop the term along with , the leading order terms give

\[b e^{i \delta}=-f / 2 m(\varepsilon-i \lambda) \omega_{0}\]

so the response, the dependence of amplitude \(b\) on driving frequency \(\Omega=\omega_{0}+\varepsilon\) is to this accuracy

\[b=\frac{f}{2 m \omega_{0} \sqrt{\left(\gamma-\omega_{0}\right)^{2}+\lambda^{2}}}=\frac{f}{2 m \omega_{0} \sqrt{\varepsilon^{2}+\lambda^{2}}}\]

(Note also that the resonant frequency is itself lowered by the damping, another second-order effect we’ll ignore.)

The rate of absorption of energy equals the frictional loss. The friction force \(2 \lambda m \dot{x}\) on the mass moving at \(\dot{x}\) is doing work at an average rate:

\[2 \lambda m \overline{x^{2}}=\lambda m b^{2} \gamma^{2}\]

The half width of the resonance curve as a function of \(\gamma\) is given by the damping. The total area under the curve is independent of damping.

For future use, we’ll write the above equation for the amplitude \(b\) in terms of deviation \(\varepsilon\) from the resonant frequency \(\omega_{0}\).

\[b^{2}\left(\varepsilon^{2}+\lambda^{2}\right)=\frac{f^{2}}{4 m^{2} \omega_{0}^{2}}, \quad \varepsilon=\gamma-\omega_{0}\]