Physics
Forced Oscillation
Quick fact
The Tacoma Narrows Bridge collapse in 1940 was a dramatic example of forced oscillation: wind acted as a periodic driving force that matched the bridge's natural frequency, causing destructive resonance.
Why this is interesting
Have you ever noticed how pushing a swing at just the right moments makes it go higher, while pushing at the wrong times barely moves it? That is the essence of forced oscillation.
Read the full explanation
Understanding Forced Oscillation
Imagine a child on a swing. If you give a single push, the swing oscillates at its own natural frequency, gradually slowing due to friction. Now imagine you push the swing repeatedly, once per cycle. The swing is forced to move at the frequency of your pushes, not its natural frequency. This is forced oscillation. The size of the swing's motion (amplitude) depends on how close your pushing frequency is to the swing's natural frequency. If you push in sync (at the natural frequency), each push adds energy and the amplitude grows—this is resonance. If you push at a very different frequency, some pushes oppose the motion and the amplitude stays small. In real systems, damping (like air resistance) always reduces the amplitude and slightly shifts the resonance peak.
A deeper explanation
In forced oscillation, an external periodic force F(t) = F₀ cos(ωt) acts on a damped harmonic oscillator. The equation of motion is: m d²x/dt² + b dx/dt + kx = F₀ cos(ωt). The system eventually reaches a steady state where it oscillates at the driving frequency ω, not its natural frequency ω₀ = √(k/m). The amplitude A(ω) = F₀ / √((k - mω²)² + (bω)²). Resonance occurs when ω ≈ ω₀, leading to maximum amplitude, limited only by damping b. The phase difference between driving force and displacement shifts from 0 to π as ω passes through ω₀. This concept explains why soldiers break step when crossing a bridge (to avoid resonant forcing), how radio tuners select specific frequencies, and why energy is efficiently transferred in musical instrument soundboards. Without damping, resonance would produce infinite amplitude—but all real systems have some damping, which broadens the resonance curve.