Damped, Forced Oscillations, and Resonance
In damped oscillations, a resistive force such as friction or drag removes energy so that the amplitude decreases with time; in forced oscillations, a periodic driving force makes the system oscillate at the driving frequency, and resonance occurs when the driving frequency matches the natural frequency f0 = (1/2π)√(k/m), producing a maximum amplitude. In brief: damping can be light (many oscillations before stopping), critical (the fastest return to equilibrium without oscillating) or heavy (slow return, no oscillation). Light damping gives a sharp, tall resonance peak, and heavy damping a low, broad one. Resonance is useful in musical instruments and radio tuners, and dangerous in bridges and buildings.
A car’s shock absorbers, a child on a swing pushed at the right time and a wine glass shattered by sound are examples of damped and forced oscillation. This guide describes the types of damping, forced oscillations and resonance, compares the damping regimes in a table, works through numerical examples of decay and the quality factor, and corrects common mistakes.
What is damped oscillation?
In real systems, friction, air resistance or internal friction removes mechanical energy, and the oscillation dies down. For a damping force proportional to velocity, F = −bv, the amplitude decays exponentially: A(t) = A0e−bt/2m. The frequency of oscillation is slightly lower than the natural frequency. Three regimes:
- Underdamped (light damping): the system oscillates with gradually decreasing amplitude. Each swing is a nearly constant fraction of the previous one.
- Critically damped: the system returns to equilibrium in the shortest time without overshooting. Used in car suspensions, door closers, and meter pointers.
- Overdamped (heavy damping): the system returns slowly to equilibrium without oscillating (for example, a pendulum in thick oil).
The quality factor Q measures how lightly damped a system is: Q ≈ 2π × (energy stored ÷ energy lost per cycle). A high Q means that the oscillation lasts for many cycles. A tuning fork has a Q of thousands, and a quartz crystal in a watch has a Q of about 104 to 106.
What are forced oscillations?
A periodic external force of frequency f drives the system. After a short transient, the system oscillates at the driving frequency f (not its own), with an amplitude that depends on how close f is to the natural frequency f0, and on the damping. The amplitude is small when f is far from f0, and increases sharply as f approaches f0. The phase of the oscillation relative to the driving force changes from 0 (in phase, at low frequency) through π/2 (at resonance) to π (antiphase, at high frequency). Examples: a car body vibrating over a bumpy road, an engine vibrating a panel, and a building shaken by an earthquake.
What is resonance?
Resonance occurs when the driving frequency equals the natural frequency of the system. The driving force is then always in step with the velocity, so that it does positive work during the whole cycle and the energy supplied increases the amplitude, until the energy supplied per cycle equals the energy lost to damping. Light damping gives a tall, narrow resonance peak; heavy damping a low, broad one. Examples:
- Useful: pushing a swing at its natural frequency; musical instruments (the sound box of a guitar or violin, air columns in pipes); radio and TV tuning circuits (LC resonance); MRI (nuclear magnetic resonance); lasers (resonant cavities).
- Harmful: the collapse of the Tacoma Narrows Bridge (1940), which was caused by wind-driven aeroelastic flutter, often presented as resonance; the Millennium Bridge in London (2000), whose lateral swaying was increased by pedestrians walking in step with it, and which was fitted with dampers; vibrations of engines, rattling of loose parts in cars; soldiers break step when marching across a bridge. Buildings in earthquake zones are designed with tuned mass dampers (such as the 660-tonne pendulum in Taipei 101) to prevent resonance.
Damping regimes compared
| Regime | Behavior after displacement | Example |
|---|---|---|
| Undamped | Oscillates forever with a constant amplitude | Ideal frictionless pendulum |
| Underdamped | Oscillates with decaying amplitude | Guitar string, playground swing |
| Critically damped | Returns fastest, with no oscillation | Car shock absorbers, door closers |
| Overdamped | Returns slowly, with no oscillation | Pendulum in heavy oil |
Worked example: decay, resonance and the Q factor
- Natural frequency. A 0.50 kg mass on a spring with k = 200 N/m: ω0 = √(k ÷ m) = √400 = 20 rad/s, f0 = 20 ÷ 2π = 3.18 Hz. Driving the system at 3.2 Hz gives a large response; at 1.0 Hz or 10 Hz, the response is small.
- Decay of amplitude. If the amplitude falls by 5 percent each cycle (a factor of 0.95), after n cycles A = A0(0.95)n. The amplitude halves when n = ln 0.5 ÷ ln 0.95 = 13.5 cycles, which takes 13.5 × 0.314 s = 4.2 s.
- Energy lost per cycle. Energy is proportional to the square of the amplitude, so each cycle the energy decreases by a factor 0.952 = 0.9025, a loss of 9.75 percent per cycle.
- Quality factor. Q ≈ 2π ÷ (fractional energy loss per cycle) = 2π ÷ 0.0975 = 64. A system with a Q of 64 has a resonance peak with a width of about f0 ÷ Q = 3.18 ÷ 64 = 0.05 Hz.
- Pushing a swing. A swing has a period of 3.0 s. Pushing once every 3.0 s (f = 0.33 Hz) increases the amplitude, with each push adding energy. Pushing every 2.0 s or 4.5 s mostly opposes the motion and the amplitude stays small.
How was resonance understood?
Galileo noticed in the 17th century that a pendulum can be set in motion by small pushes in time with its swing. Hermann von Helmholtz used resonators to analyze the harmonics of sound in the 1860s. Heinrich Hertz detected radio waves with a resonant loop in 1887, and radio tuning circuits are based on electrical resonance. The study of the Tacoma Narrows Bridge failure led to the aerodynamic testing of bridges in wind tunnels.
Which mistakes are common?
Mistake 1: Thinking that a forced system oscillates at its natural frequency
In steady state it oscillates at the driving frequency. The natural frequency determines the response amplitude.
Mistake 2: Saying that damping removes the resonance
Damping reduces and broadens the peak but does not remove it, unless it is very heavy.
Mistake 3: Believing that the amplitude at resonance becomes infinite
It is limited by damping. Without damping, the amplitude would grow until something fails.
Mistake 4: Confusing critical damping with overdamping
Critical damping is the fastest return to equilibrium. Overdamping is slower.