16.8 Forced Oscillations and Resonance

Forced Oscillations and Resonance

Every oscillating system has a natural frequency it "prefers" to vibrate at. When an external force drives that system at or near its natural frequency, the amplitude can grow dramatically. This effect is called resonance, and it shows up everywhere, from playground swings to skyscrapers swaying in the wind. Understanding how driving forces, natural frequencies, and damping interact is essential for predicting when resonance helps (musical instruments) and when it's dangerous (bridge collapses).

Resonance and Natural Frequency

A natural frequency is the frequency at which a system tends to oscillate when you disturb it and let it go. Pull a pendulum to one side and release it, and it swings back and forth at its natural frequency. Pluck a guitar string, and it vibrates at its natural frequency too.

Resonance occurs when an external driving force is applied at a frequency that matches the system's natural frequency. At resonance:

• Energy transfer from the driving force to the system is maximally efficient.
• The driving force adds energy in sync with the oscillation each cycle, so amplitude builds over time.
• Without any damping, the amplitude would theoretically grow without limit (though in real systems, damping and material limits always cap it).

A few familiar examples:

• Pushing a child on a swing. If you push at the same rate the swing naturally moves, each push adds energy and the swing goes higher. Push at the wrong rate, and you fight the motion.
• Musical instruments. A guitar body is shaped so that its air cavity resonates at frequencies matching the strings, amplifying the sound.
• Electrical circuits. RLC circuits resonate at a specific frequency determined by their inductance and capacitance, which is how radios tune to a station.

One more term to know: a harmonic oscillator is any system where the restoring force is proportional to displacement from equilibrium. Springs and pendulums (for small angles) are classic examples, and they form the foundation for analyzing resonance.

Damping Effects on Oscillation Amplitude

Damping is the loss of energy from an oscillating system due to friction, air resistance, or other dissipative forces. It's what makes a pendulum eventually stop swinging.

Damping has a direct effect on forced oscillations:

• Greater damping means smaller oscillation amplitudes overall.
• At resonance specifically, heavier damping lowers the peak amplitude. Light damping produces a tall, sharp resonance peak; heavy damping produces a broad, low one.
• Without damping, amplitude at resonance would grow indefinitely. Damping is what keeps real systems from tearing themselves apart.

There are three damping regimes, and you should be able to distinguish them:

1. Underdamped — The system oscillates with gradually decreasing amplitude. A pendulum swinging in air is underdamped; it completes many cycles before stopping.
2. Critically damped — The system returns to equilibrium as fast as possible without oscillating past it. A well-designed door closer does this: the door swings shut smoothly without bouncing.
3. Overdamped — The system returns to equilibrium slowly, without oscillating, because damping forces are so strong. Think of pushing a heavy object through thick honey.

The quality factor (often written as $Q$) is a dimensionless number that describes how underdamped an oscillator is. A high $Q$ means the system loses very little energy per cycle and has a sharp resonance peak. A low $Q$ means heavy damping and a broad, flattened peak.

Forced Oscillation Characteristics

When an external periodic force drives an oscillator, three quantities matter beyond amplitude:

• Forced (driving) frequency — The frequency of the external force, which may or may not match the natural frequency. The system oscillates at the driving frequency, not its natural frequency, once it reaches steady state.
• Phase shift — The oscillation doesn't necessarily move in perfect sync with the driving force. At low driving frequencies, the system nearly follows the force. At resonance, the oscillation lags the force by $90°$. Well above resonance, the lag approaches $180°$, meaning the system moves opposite to the applied force.
• Mechanical impedance — The ratio of the applied force to the resulting velocity. It's the mechanical analog of electrical impedance. High impedance means the system resists being driven; at resonance, impedance is minimized (for the component related to mass and stiffness), which is why amplitude peaks.

Applications of Forced Oscillations

Mechanical systems:

• Engines and motors generate periodic forces that can excite resonance in nearby components. Engineers use balancing and damping to keep vibrations under control.
• Bridges and buildings must be designed so their natural frequencies don't align with common wind or seismic frequencies. Taipei 101, for example, contains a massive tuned mass damper to counteract wind-driven oscillations.
• Vehicle shock absorbers are deliberately designed to be near critically damped, so bumps from the road decay quickly without causing the car to bounce repeatedly.

Biological systems:

• Human hearing relies on resonance. The cochlea in your inner ear contains structures that resonate at different frequencies along its length, allowing you to distinguish pitches from about 20 Hz to 20,000 Hz.
• The cardiovascular system can experience resonance effects in the aorta, which in extreme cases may contribute to aneurysm formation.
• Trees and plants have natural sway frequencies. Strong, sustained winds at the right frequency can cause resonance that leads to structural damage or uprooting.

Resonance disasters and how to prevent them:

Two well-known cases illustrate what happens when resonance goes unchecked:

1. Tacoma Narrows Bridge (1940) — Wind excited oscillations that grew until the bridge tore itself apart. (The exact mechanism involved aeroelastic flutter, but the core lesson is about uncontrolled energy input at a structural frequency.)
2. Millennium Bridge, London (2000) — Pedestrians unconsciously synchronized their footsteps with the bridge's lateral sway, feeding energy into the oscillation and making it worse. The bridge had to be closed and retrofitted.

Engineers use several strategies to prevent resonance disasters:

• Material selection and structural design that place natural frequencies away from expected driving frequencies.
• Tuned mass dampers — large masses on springs or pendulums inside a structure that oscillate out of phase with the building, absorbing energy. Taipei 101 and the Burj Khalifa both use these.
• Active control systems that monitor vibrations in real time and apply counteracting forces, such as magnetorheological dampers whose resistance changes with an applied magnetic field.