Forced Oscillation & Resonance Reference Cheat Sheet

Key Facts

• For an ideal mass spring oscillator, the natural angular frequency is $\omega_0 = \sqrt{\frac{k}{m}}$ and the frequency is $f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$.
• A driven damped oscillator is modeled by $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)$.
• The steady state displacement of a driven oscillator can be written as $x(t) = A\cos(\omega t - \phi)$.
• The amplitude response is $A = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (b\omega)^2}}$ for a sinusoidal driving force.
• Resonance occurs when the driving angular frequency is near the natural angular frequency, so $\omega \approx \omega_0$ for light damping.
• Damping reduces the maximum amplitude and spreads the resonance peak over a wider range of frequencies.
• The quality factor can be estimated by $Q = \frac{\omega_0}{\Delta \omega}$, where $\Delta \omega$ is the full width of the resonance peak at half maximum power.
• For weak damping, the average power absorbed is greatest near resonance because the driving force transfers energy efficiently to the oscillator.

Vocabulary

Forced oscillation

A motion in which an external periodic force makes a system oscillate at the driving frequency.

Natural frequency

The frequency at which a system oscillates on its own after being disturbed, written as $f_0$ or $\omega_0$.

Driving frequency

The frequency of the external periodic force applied to an oscillator, written as $f$ or $\omega$.

Resonance

The condition in which a driven oscillator reaches a large amplitude because the driving frequency is close to its natural frequency.

Damping

The loss of mechanical energy from an oscillator due to forces such as friction, air resistance, or internal resistance.

Quality factor

A measure of how sharp a resonance is, often written as $Q = \frac{\omega_0}{\Delta \omega}$.