Forced Oscillations
Amplitude vs driving frequency curve.
Key Notes
Forced oscillation: a driving force F = F₀·cos(ω_d·t) drives a damped oscillator at frequency ω_d (different from natural ω₀).
Steady-state response: x(t) = A_d·cos(ω_d·t − φ), where A_d depends on ω_d.
Amplitude A_d depends on (i) driving force F₀, (ii) ω_d vs ω₀, (iii) damping b.
Maximum amplitude occurs at RESONANCE ω_d ≈ ω₀ — sharp peak for small damping.
Phase φ between drive and response: 0 at low ω_d, π/2 at resonance, π at high ω_d.
Energy is continuously supplied by drive and dissipated by damping at steady state.
Transient response (initial conditions) dies out exponentially — only steady-state remains.
Examples: child on swing, AC circuit driven by source, building swaying in wind, tuning circuits.
Formulas
Driving force
Sinusoidal at driving frequency ω_d.
Steady-state amplitude
Peaks near ω_d ≈ ω₀ when b small.
Phase lag
φ = 0 at ω_d ≪ ω₀; π/2 at resonance; π at ω_d ≫ ω₀.
Resonance frequency (damped)
For light damping, very close to natural frequency.
Important Points
At RESONANCE: amplitude is maximum, can be much larger than F₀/k.
Far from resonance, response amplitude is small.
Phase lag: drive leads response by 0 (low ω), π/2 (resonance), π (high ω).
Low damping ⇒ sharp resonance, high amplitude. High damping ⇒ broad, low resonance.
Tacoma Narrows Bridge (1940 collapse): wind-induced oscillation at resonance frequency.
Resonance frequency in damped systems is slightly LESS than ω₀ — for very light damping, the difference is negligible.
Forced Oscillations notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.