Forced oscillations and resonance

If we apply an external periodic force $F_\text{est}(t) = F_0\cos(\omega t)$ to an oscillator, with or without damping, the system enters a forced regime. After an initial transient, the motion becomes oscillatory at the driving frequency (not the natural one):

$x(t) = A(\omega)\cos(\omega t - \delta)$

The steady-state amplitude depends on the distance between the driving frequency $\omega$ and the natural frequency $\omega_0$, and it is larger the closer the two are:

$\ev{A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}}}$

Principle — Resonance

The amplitude $A(\omega)$ has a peak when $\omega \approx \omega_0$. This is the phenomenon of resonance: a driving force at the system’s natural frequency enormously amplifies its amplitude, limited only by damping. As $\zeta\to 0$ the amplitude at the peak tends to infinity (linear catastrophe) (Pain 2005). See Riferimenti bibliografici.

Resonance curves for three values of damping. The smaller $\zeta$ is, the higher and narrower the peak around $\omega = \omega_0$.

Resonance in nature and technology

Resonance is the principle behind:

• the Tacoma Narrows Bridge, which collapsed in 1940 under a wind that excited a torsional mode at about $0{,}2$ Hz;
• the soundbox of a guitar or violin, tuned to the useful frequencies;
• the microwave oven, tuned to the vibrational mode of water at about $2{,}45$ GHz;
• nuclear magnetic resonance (MRI) in medicine;
• the $LC$ circuit of analogue radios, where the link with the mass-spring system will become explicit in the chapter on induction.

The swing analogy

Resonance means “pushing at the right rhythm”: if you push a swing in phase with its natural frequency, every push adds energy and the amplitude grows without bound. Pushing out of time, instead, sometimes adds and sometimes removes energy, and you get nowhere.

▶ What If Swings Had Springs: Autoparametric Resonance — Steve Mould

Links

Topics: Oscillations and harmonic motion Concepts: Resonance · Damped oscillations Objects: Spring

Related exercises: Problem — Natural frequency of a skyscraper · Problem — Damping and isochronism · Worked exercise — Mass, spring and damping