Damped Oscillations and Q-Factor
On this page
Direct answer
Amplitude that decays exponentially is the signature of damping: for an oscillator m x'' + b x' + kx = 0, the damping constant β = b/2m makes the amplitude follow A = A0e^−βt and slows the frequency to ωd = √(ω0² − β²), where ω0 = √(k/m) is the undamped natural frequency. Damping is underdamped (β < ω0, oscillation with shrinking amplitude), critical (β = ω0, fastest non-oscillatory return) or overdamped (β > ω0, slow creep back). The quality factor Q = ω0/(2β) measures how many radians the oscillator survives — energy decays as e^−2βt — and equals ω0/Δω, the ratio of resonance frequency to resonance bandwidth. The examiner's favourite subtlety: under forced vibration, displacement amplitude peaks at ω_res = √(ω0² − 2β²), slightly below ω0, while velocity resonance sits exactly at ω0.
What you must remember
- Amplitude decay: A(t) = A0e^−βt with β = b/2m; time to fall to 1/e of amplitude is 1/β, and energy (which goes as A²) decays twice as fast, as e^−2βt.
- Damped frequency: ωd = √(ω0² − β²) — always below the natural frequency; for weak damping ωd ≈ ω0.
- Three regimes: underdamped β < ω0 (oscillates), critically damped β = ω0 (returns to equilibrium fastest without oscillating — the goal for vehicle shock absorbers and galvanometer pointers), overdamped β > ω0 (sluggish return).
- Quality factor: Q = ω0/(2β) = 2π × (energy stored)/(energy lost per cycle); high Q means slow decay and a sharp, tall resonance peak.
- Bandwidth: Δω = ω0/Q between the half-power points; sharper resonance means better frequency selectivity (tuning circuits, musical instruments).
- Resonance split: displacement amplitude resonates at ω_res = √(ω0² − 2β²) < ω0; velocity (and average power) resonance occurs exactly at ω0 — a distinction JEE Advanced has tested directly.
- Logarithmic decrement: λ = βT_d, the natural log of successive amplitude ratios, the standard way damping is measured experimentally.
Numbers worth knowing
Give an oscillator ω0 = 10 rad/s and β = 2 s⁻¹. It is underdamped (2 < 10), so it still oscillates, but at ωd = √(100 − 4) = 9.80 rad/s — measurably slower than natural. The quality factor is Q = 10/4 = 2.5, the amplitude falls to 1/e in half a second, and the resonance bandwidth is Δω = ω0/Q = 4 rad/s: a broad, humble peak. Displacement resonance would occur at √(100 − 8) = 9.59 rad/s while velocity resonance sits at 10 — the three frequencies 9.59, 9.80 and 10 are all different, and a well-set multiple-correct question asks you to rank them.
Now push β down to 0.5: ωd = √(100 − 0.25) = 9.99 rad/s, Q = 10, Δω = 1 rad/s, amplitude survives ten full swings before decaying by 1/e — the peak sharpens and shifts back toward ω0. This pair of computations is the entire intuition of the chapter: damping simultaneously lowers the frequency, lowers the peak, widens the band and hastens the death of free oscillations; every resonance curve drawn in the paper encodes all four effects at once, and reading a curve for "which has more damping" is a two-mark recognition task every year.
How JEE frames damping
The most valuable fact, because it is counterintuitive: displacement resonance frequency lies below the natural frequency (the 2β² under the square root), while velocity resonance is exactly at ω0; options offering ω0 for displacement resonance are bait. Second, critical damping is the fastest return without overshoot — not the slowest, and not the most damped; students who equate "critical" with "extreme" choose overdamped and lose the mark. Third, Q defined through energy (2π E/ΔE per cycle) and through bandwidth (ω0/Δω) is the same quantity; questions hop between definitions to check understanding rather than memorisation. Fourth, the e^−2βt energy decay versus e^−βt amplitude decay pairing appears in statement-based format. Main sticks to regime identification and amplitude ratios; Advanced builds the amplitude-resonance shift, the logarithmic decrement, or the shock-absorber design logic into longer questions.
Frequently asked questions
How does damping change the frequency of an oscillator?
The oscillation frequency drops to ωd = √(ω0² − β²); for weak damping the shift is negligible, but it grows as damping approaches critical.
What is critical damping and where is it desired?
The condition β = ω0, where the system returns to equilibrium in the shortest time without oscillating — the design target for vehicle suspensions and dead-beat galvanometer pointers.
What does the quality factor Q measure?
The sharpness of resonance: Q = ω0/(2β) = ω0/Δω, equivalently 2π times the ratio of stored energy to energy dissipated per cycle.
At what frequency is displacement amplitude maximum in forced oscillations?
At ω_res = √(ω0² − 2β²), slightly below the natural frequency — whereas velocity and power resonance occur exactly at ω0.
How is damping measured from a recorded oscillation trace?
Through the logarithmic decrement λ = βT_d, the natural log of the ratio of two successive amplitudes one period apart.