# Resonance

## Alexei Gilchrist

### 1 Introduction

On a long enough time scale the solution to the driven and damped oscillator (driven-damped-oscillator) becomes dominated by what is known as the “steady state” solution, which looks like

If \(\gamma\) was zero and we drove the oscillator so that \(\omega_f=\omega_0\), \(A_p\) would become infinite! Something interesting is happening here and this is *resonance*.

### 2 Amplitude resonance

First consider the limits of \(A_p(\omega_f)\) as a functions of \(\omega_f\). When \(\omega_f\rightarrow\infty\) then \(A_p\rightarrow 0\) and at the other limit \(A_p(0) = f_0/\omega_0^2\). In between, the amplitude peaks at some value of \(\omega_f\).

We can find where the maximum occurs by solving \(dA_p/d\omega_f=0\) for \(\omega_f\),

*(i)*\(\omega_f=0\),

*(ii)*\(\omega_f=-\sqrt{\omega_0^2-\gamma^2/2}\), and

*(iii)*\(\omega_f=\sqrt{\omega_0^2-\gamma^2/2}\). The first solution is trivial, the second unphysical (negative frequencies), which leaves the third so the resonance angular frequency is

How big will the response be? Now we know at which frequency resonance occurs, we can evaluate \(A_p\) at that point, \(A_p^\mathrm{max} = A_p(\omega_r) = f_0 / \gamma \omega_d\). This makes it clear that the response becomes large when \(\gamma\) becomes small! In fact, for weak damping \(\omega_d\) is approximately \(\omega_0\), i.e. \(\omega_d=\sqrt{\omega_0^2-\gamma^2/4}\approx \omega_0\), and the ratio of the amplitude driven at resonance to the amplitude at zero frequency is given by the \(Q\) of the oscillator:

The phase shift

This behaviour can be quickly verified with a simple experiment—hang a small weight onto the end of a rubber band. If you drive the weight sinusoidally very slowly you will see that the response is some fixed amplitude and in-phase. If you increase the driving frequency, at some value you will hit resonance and it will bob up and down like crazy. Above that, the weight settles into periodic motion that is out of phase with your driving. If you drive it as fast as you can (keeping the motion even) then the weight will hardly move despite the driving force, a result of \(A_p\rightarrow 0\).

The following simulation shows how the resonance peak broadens out with increasing damping.

### 3 Velocity resonance

It may be surprising to discover that the velocity has a different resonance to the amplitude but this is indeed the case! The velocity in the steady state is given by

In the limit of weak damping both these resonances, and that of the energy discussed below, become \(\omega_0\).

### 4 The steady state energy

We already have the position \(x(t)\), so we can get the velocity by differentiating: \(\dot{x} = -\omega_f A_p\sin(\omega_f + \phi_p)\), consequently the kinetic (\(E_k\)), potential (\(E_p\)), and the total mechanical energy (\(E_m\)) is

If we examine the mechanical energy of the oscillator averaged over a cycle we find

*at*resonance the total mechanical energy is constant even within a cycle and the system behaves as an undamped oscillator.

The simulation above shows that the energy in a driven oscillator depends strongly on the driving frequency. Examining this more closely we can expand \(\langle E_m \rangle_c\) to show the dependence on \(\omega_f\) explicitly. Remember that \(A_p\) depends on \(\omega_f\) too.