6.5 Forced Oscillations

A forced oscillator is an oscillator subject to a periodic external force.

F_{\text{net}}=-kx-bv+F_{\text{drive}}.

(6.68)

Consider a sinusoidal driving force $F_{\text{drive}}=F_{0}\cos(\omega_{\text{dr}}t)$ . Then the equation of motion becomes

m\frac{\operatorname{\mathrm{d}}\!^{2}x}{\operatorname{\mathrm{d}}\!t^{2}}+b% \frac{\operatorname{\mathrm{d}}\!x}{\operatorname{\mathrm{d}}\!t}+kx=F_{0}\cos% (\omega_{\text{dr}}t).

(6.69)

When we solve this equation, what we find is that, when damping is small, there is a huge build-up of energy when the system is driven at its resonant frequency $\omega_{0}$ . This phenomenon is called resonance. Mathematically, the condition for resonance is

\omega_{\text{drive}}=\sqrt{\omega_{0}^{2}-\frac{b^{2}}{2m^{2}}}=\sqrt{\omega_% {0}^{2}-2\gamma^{2}}.

(6.70)

We call this $\omega_{\text{res}}$ . Note that this is not the damped frequency $\omega_{d}$ from above. We see that when damping is weak, the condition for resonance becomes $\omega_{\text{drive}}\approx\omega_{0}$ .

To find the resonant frequency of a system, there are two general methods we can use. There is the “impulse method”, where we excite all frequencies in the system at once. The system will then resonate at the resonant frequency and the vibrations at other frequencies will decay away (we will see why shortly). An example of this is striking a bell. The other method is the “frequency-scan method”, where we use a low-ampltidue signal and scan through frequencies until we hit resonance, when energy will rapidly build up.

6.5.1 Underdamped Driven Oscillations

For the case of underdamping ( $\gamma<\omega_{0}$ ), the general solution is a sum of a transient (decaying) oscillation at the damped frequency and a steady state (not decaying) oscillation at the driven frequency.

x(t)=A_{\text{decay}}e^{-\gamma t}\cos(\omega_{d}t+\phi_{\text{decay}})+A_{% \text{steady}}\cos(\omega_{\text{drive}}t-\phi_{\text{steady}}).

(6.71)

The ampltiude and phase of the steady state oscillation depend on the driven frequency and the resonant frequency. These dependencies have the form

A_{\text{steady}}=\frac{F_{0}/m}{\sqrt{(\omega_{\text{dr}}^{2}-\omega_{0}^{2})% ^{2}+\left(\frac{b}{m}\right)^{2}\omega_{\text{dr}}^{2}}},\quad\phi_{\text{% steady}}=\tan^{-1}\left(\frac{\left(\frac{b}{m}\right)\omega_{\text{drive}}}{% \omega_{0}^{2}-\omega_{\text{drive}}^{2}}\right)+\phi_{0}

(6.72)

$A_{\text{steady}}$ has a maximum at $\omega_{\text{res}}$ , which is the resonance phenomenon we have been discussing. Note that we now have three different “resonance” frequencies for oscillators. $\omega_{0}$ for when we have an undamped, undriven oscillator (SHM), $\omega_{d}$ for a damped oscillator, and $\omega_{\text{res}}$ for a driven oscillator. We see that at low driving frequencies, the ampltide of the steady state is small (but nonzero) but the oscillation is in-phase with the driving force. For high driving frequencies, the amplitude goes to zero, and the oscillation is in anti-phase with the driving force. At resonance, the phase of the steady state is $\pi/2$ , which means the driving force is in-phase with the velocity. This makes sense because then we have a constant power input to the system ( $P=\boldsymbol{F}\cdot\boldsymbol{v}$ ).

For a driven oscillator, we find that the $Q$ -factor is

Q=\frac{\omega_{\text{res}}}{\text{FWHM of energy curve}}\approx\omega_{\text{% res}}\frac{m}{b}.

(6.73)

If damping is low, we can say $Q\approx\omega_{0}\frac{m}{b}$ and write

A_{\text{steady}}\approx\frac{F_{0}(\omega_{0}/\omega_{\text{drive}})}{k\sqrt{% \left(\frac{\omega_{0}}{\omega_{\text{drive}}}-\frac{\omega_{\text{drive}}}{% \omega_{0}}\right)^{2}+\frac{1}{Q^{2}}}}.

(6.74)

So for $Q\gg 1$ , we get the following cases. For $\omega_{\text{drive}}\ll\omega_{\text{res}}$ , $A_{\text{steady}}=F_{0}/k$ . For $\omega_{\text{drive}}=\omega_{\text{res}}$ , we get $A_{\text{steady}}=QF_{0}/k$ . For $\omega_{\text{drive}}\gg\omega_{\text{res}}$ , we have $A_{\text{steady}}=-\frac{\omega_{0}}{\omega_{\text{drive}}}F_{0}/k$ .