Forced Oscillations and Resonance: A Complete Guide

A forced oscillation happens when a periodic outside force keeps pushing a vibrating system — like pushing a child on a swing at regular intervals, or an unbalanced washing machine shaking the floor. Unlike free vibration, which dies out from damping, a forced system settles into a steady oscillation at the driving frequency, powered continuously by the external push.

This is one of physics' most practical ideas because resonance — the dramatic growth when you push at the natural frequency — can be wonderfully useful or dangerously destructive. Engineers exploit it in radios and instruments, and fight it in bridges, buildings, and aircraft.

Key Concepts

• Driving force F = F_0 cos(ωt): amplitude F_0 (N), driving frequency ω (rad/s).
• Natural frequency ω_0 = √(k/m): the frequency the system prefers on its own.
• Resonance: when ω = ω_0, each push lands in step with the motion, energy piles up, and the amplitude reaches its maximum.
• Transient vs steady state: the motion starts messy, but damping kills the natural part, leaving a clean oscillation at the driving frequency.
• Phase: below resonance the system moves in step with the push; above resonance it moves opposite; at resonance it lags by a quarter cycle.

Formulas

• Steady-state amplitude: A = F_0 / √(m²(ω_0² - ω²)² + b²ω²). This single formula draws the whole resonance curve.
• Amplitude at resonance (ω = ω_0): A_res = F_0 / (b ω_0). Notice mass and stiffness vanish from it — only the push and the damping matter at the peak.
• Units: F_0 in N, b in kg/s, ω in rad/s, A in metres.
• Assumptions and limits: linear spring, viscous damping proportional to velocity, and amplitudes small enough that nothing bends or breaks. With zero damping the formula predicts infinite amplitude — in reality the system breaks or becomes non-linear first.

Worked Example

A 2 kg machine part on a 200 N/m mount is shaken by a 5 N force, with damping b = 1 kg/s.

1. Natural frequency: ω_0 = √(200/2) = √100 = 10 rad/s, i.e. f_0 = 10/2π = 1.59 Hz.
2. At resonance (drive at 10 rad/s): A = F_0/(b ω_0) = 5/(1 x 10) = 0.5 m — a huge shake from a mere 5 N push.
3. Off resonance (drive at 12 rad/s): denominator = √(4 x (100-144)² + 1 x 144) = √(7744 + 144) = √7888 = 88.8, so A = 5/88.8 = 0.056 m.
4. Just a 20% shift in driving frequency shrinks the vibration nearly nine times — that is why staying off resonance matters.

Common Misconceptions

• "Resonance means infinite vibration." Only in the fantasy of zero damping. Real damping always caps the peak at F_0/(bω_0) — but that capped value can still be big enough to destroy things.
• "Pushing harder is the way to get a big swing." Timing beats strength: small pushes in rhythm (at ω_0) build a far bigger swing than large random pushes.
• "Tacoma Narrows was pure resonance." Mostly it was aeroelastic flutter — wind vortices feeding the twist — but the lesson stands: periodic forcing near a natural frequency can wreck structures.

Real-World Applications

• Radio tuning: an LC circuit resonates at one station's frequency, rejecting the rest.
• Playground swings: pumping your legs at the swing's natural frequency is resonance in action.
• Musical instruments: soundboards and air columns resonate with the string or reed to amplify the tone.
• Structural design: engineers keep bridges' and buildings' natural frequencies far from wind, traffic, and earthquake frequencies.

In short: a forced oscillator vibrates at the driving frequency, and at resonance the amplitude peaks at F_0/(bω_0). Small well-timed pushes beat big mistimed ones — behind both beautiful music and collapsed bridges.