Damped & Forced Oscillations, Resonance

Flick a wine glass and it sings for seconds — thousands of vibrations before the note dies. Push a screen door fitted with a hydraulic closer and it swings shut once, no wobble, no bounce. Drive a car over a pothole and the body dips, recovers, and settles in under a second. All three are the same system — a mass, a restoring force, and friction — living in three different regimes of one physics. The only thing separating a singing glass from a door closer is how fast friction eats the energy compared with how fast the oscillation wants to cycle. Every oscillator you have met so far was a beautiful lie: the frictionless spring–block and pendulum keep their energy forever. Real oscillators leak. This module tells the honest story in three acts. Damped oscillations: the amplitude dies away — and it dies in a very particular pattern, losing the same fraction every cycle, not the same amount. Forced oscillations: push a system rhythmically and, after a brief argument, it surrenders and vibrates at your frequency, not its own. And resonance: when your pushing rhythm matches the system's natural rhythm, tiny pushes stack into enormous swings — the effect that lets a child pump a swing to frightening heights, lets a radio pluck one station out of a thousand, and closed London's Millennium Bridge two days after it opened. At HS level this chapter is asked mostly qualitatively — sketch the decay, read the resonance curve, name who sets the frequency — with a few clean numbers on top. Those few marks are among the easiest in the paper if you carry the right three or four sentences in your head. This module hands them to you.

Key takeaways

Damping is a resistive force (drag, friction) that drains an oscillator's energy: the amplitude decays by equal fractions in equal times — a geometric (exponential) fade, not a straight line.
Because energy $\propto A^2$ , energy dies twice as fast as amplitude: when $A$ has halved, $E$ is down to one quarter.
Light damping barely changes the frequency (slightly lower); heavy damping kills oscillation entirely; critical damping gives the fastest return to rest with no overshoot — what car suspensions and door closers are tuned for.
A forced oscillator in steady state vibrates at the driver's frequency, whatever its own natural frequency $f_0$ is; the natural frequency only decides how large the response is.
Resonance: the response amplitude peaks when the driving frequency matches the natural frequency ( $f_d \approx f_0$ ); lighter damping makes the peak taller and sharper.
Key idea: friction makes every free oscillation fade by equal fractions per cycle; a rhythmic driver takes command of the frequency — and when the driver's rhythm hits the natural one, amplitude piles up until damping alone calls the ceiling.

The core insight

Damping drains an oscillator geometrically (equal fractions of amplitude per cycle, energy falling as the square); a periodic driver forces steady vibration at its own frequency, with amplitude largest — and limited only by damping — when the drive matches the natural frequency: resonance.

Where this lives today

Damping and resonance are twin design disciplines now. On the damping side: every car on the road carries four tuned dampers holding the suspension near critical (worn "shocks" push it underdamped — the corner-bounce test detects the extra oscillation); skyscrapers from Taipei 101 to Shanghai Tower hang hundred-tonne tuned mass dampers that deliberately resonate with the building's sway so as to steal its energy and dump it into oil; and LIGO's mirror suspensions are engineered for the opposite extreme — the least damping (highest quality factor) achievable, because damping's flip side is noise. On the resonance side: your phone's radio circuits tune $f_0$ across channels a hundred times a second; wireless chargers couple through matched resonant coils; MRI machines drive hydrogen nuclei at their precise resonant frequency and read the reply; and earthquake engineering is largely the art of moving buildings' natural frequencies away from the ground's favourite shaking band — with the 1985 Mexico City disaster, where lakebed soil resonated at the period of 6–15 storey buildings, as the permanent cautionary tale.

The physical picture

Start with the child on the swing, because every idea in this module is visible there. Left alone after one good push, the swing is a damped oscillator: air drag and pivot friction tax each cycle, and the arc fades — noticeably shrinking by the same fraction each swing, never stopping abruptly. Now push it rhythmically. Push at random moments and you fight yourself, half your shoves braking the motion. But time your pushes to the swing's own period — one gentle shove each cycle, just as it starts moving away from you — and each deposit lands in phase with the motion. Energy stacks. The arc grows until your small pushes merely replace what friction removes each cycle: a steady state, at spectacular amplitude. You have found resonance, and also its ceiling — damping. The three characters, formally: a free oscillation runs at the system's natural frequency $f_0$ (the $\sqrt{k/m}$ or $\sqrt{g/L}$ of the previous modules) and, with damping, fades. A forced (driven) oscillation has an external periodic push and, after transients die, runs at the driver's frequency $f_d$ forever. Resonance is the special forced case $f_d \approx f_0$ where the response is maximal.

Figure

Three damping regimes — the same oscillator with light, critical and heavy friction

Damped oscillations — the honest fade

Add a resistive force that grows with speed (air drag on a pendulum bob, oil in a car damper) to any oscillator and three things happen, in order of exam importance.

First: the amplitude decays geometrically. The oscillation loses the same fraction of its amplitude every cycle — 5%, say, so the peaks run $A_0,\ 0.95A_0,\ 0.95^2A_0,\ \ldots$ — tracing the smooth dying envelope in the figure. Textbooks write this envelope as an exponential,

A(t) = A_0\,e^{-bt/2m} \qquad (1)

where: $b$ — the damping constant (drag force $= -bv$ , units kg/s); $m$ — the mass; $e \approx 2.718$ . You do not need calculus to use this: it is exactly the compound-decay pattern, "equal times, equal fractions," and every question built on it reduces to ratios. The single most useful consequence, since energy goes as amplitude squared:

E(t) = E_0\,e^{-bt/m} \qquad (2)

where: the exponent is twice as large — energy dies twice as fast as amplitude. Amplitude halved means energy quartered; amplitude down to $1/10$ means energy down to $1/100$ . Exam questions cycle through this pairing endlessly.

Second: the frequency barely budges. Light damping lowers the oscillation frequency only slightly ( $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$ , a formula to recognise rather than derive: the correction is second-order, invisible for a lightly damped system). A guitar string loses half its energy in a second yet stays perfectly in tune — decay eats amplitude, not pitch. That independence is the practical miracle that made pendulum clocks possible.

Third: enough damping kills the oscillation. Crank $b$ up and $\omega'$ slides to zero: at critical damping the system returns to equilibrium as fast as possible without crossing it — the tuning goal for car suspensions, door closers and analogue meter needles. Beyond critical (overdamping), the return gets slower again: the system oozes through treacle. "More damping = settles faster" is true only up to critical, then reverses — a favourite conceptual trap.

Derivation AEqual times, equal fractions — why the decay is geometric and energy dies twice as fast

We establish the two bookkeeping rules of damped SHM — amplitude falls by a constant ratio per cycle, and energy falls by that ratio squared — using only proportionality logic. The strategy: show that the energy a drag force eats per cycle is itself proportional to the energy present, which forces compound (geometric) decay; then translate between amplitude and energy with the $E \propto A^2$ law from the energy module. Watch how no calculus is needed: proportional loss is the whole mechanism.

Step 1. How much does drag eat per cycle? The drag force is proportional to speed, $F_d = bv$ , and the distance covered per cycle is proportional to the amplitude $A$ . Typical speeds in SHM are proportional to $\omega A$ , so the work drained per cycle scales as $F_d \times \text{distance} \sim (b\,\omega A)(A) = b\,\omega A^2$ .

Step 2. Compare with what is in the account. The stored energy is $E = \tfrac12 m \omega^2 A^2$ — also proportional to $A^2$ . So the fraction lost per cycle, $\dfrac{\Delta E}{E} \sim \dfrac{b\omega A^2}{m\omega^2 A^2} = \dfrac{b}{m\omega}$ , contains no $A$ at all: big oscillations and small ones lose the same percentage per cycle.

Step 3. Constant fractional loss is compound decay. If each cycle keeps the fraction $q$ (say $q = 0.90$ ), then after $n$ cycles $E_n = E_0\,q^n$ — a geometric sequence, which is what the smooth exponential $e^{-bt/m}$ is tracking through time. Equal times, equal fractions; never equal amounts.

Step 4. Amplitude follows as the square root. Since $E \propto A^2$ , $A_n = A_0\,q^{n/2}$ : the amplitude ratio per cycle is $\sqrt{q}$ . Run it backwards: if the amplitude keeps ratio $r$ per cycle, the energy keeps $r^2$ . Halve the amplitude, quarter the energy. Why this is the same "equal ratios" law as radioactive decay and RC discharge: in each, the loss rate is proportional to the amount present — the one assumption that manufactures exponentials, everywhere in physics. *Verified: numbers — an oscillator losing 10% of E per cycle retains $0.9^{10} = 0.35$ of its energy after 10 cycles and $0.9^{20} = 0.12$ after 20; its amplitude after 10 cycles is $\sqrt{0.35} = 0.59A_0$ ✓ (consistent: $0.59^2 = 0.35$ ). Dimensions of $b$ : N/(m/s) = kg/s ✓.*

Forced oscillations and resonance — who sets the frequency, who sets the size

Now attach a rhythmic driver — a motor shaking the spring's support, footsteps on a bridge, your hands on the swing. Two rhythms are now in the room: the system's natural $f_0$ and the driver's $f_d$ . The outcome, after a brief transient tussle, is unconditional surrender:

f_{\text{steady state}} = f_d \quad \text{(always the driver, never the natural frequency)} \qquad (3)

where: this is the most-tested sentence in the module. A driven system must repeat at the rhythm of the force sustaining it; its own $f_0$ has been demoted from "the frequency" to a preference. That preference shows up in the size of the response:

A_{\text{response}} \text{ is largest when } f_d \approx f_0 \quad \text{— resonance} \qquad (4)

where: the full response curve is plotted below. Drive far below $f_0$ and the system just placidly follows the driver (amplitude ≈ driver's own stroke); drive far above and the mass cannot keep up — the response shrivels toward zero. Near $f_0$ , each push lands in phase with the velocity, energy stacks cycle after cycle, and the amplitude climbs until damping's per-cycle tax (which grows as $A^2$ ) exactly cancels the driver's per-cycle deposit. Damping alone sets the ceiling: lighter damping, taller and sharper peak; heavier damping, a low mound. (Strictly the peak of the amplitude curve sits a whisker below $f_0$ , and the sharpness is quoted as a "quality factor" $Q$ — recognition-level facts at HS; the machinery lives in the UG deep dives.)

Figure

The resonance curve — response amplitude against driving frequency, for light and heavy damping

Resonance is everywhere exams look for it: soldiers break step on footbridges (a marching cadence near a bridge mode is a rhythmic driver — the 1831 Broughton bridge collapse made it standing orders); radio tuning turns the dial until the circuit's $f_0$ matches one broadcast frequency, resonance amplifying that station a thousandfold over its neighbours; the Millennium Bridge in London swayed alarmingly in 2000 when pedestrians' sideways footfalls locked into its 0.5 Hz lateral mode (dampers fixed it: lower, flatter peak); and MRI machines read hydrogen nuclei by driving them at their resonant frequency. One honesty note the best answers include: the famous Tacoma Narrows bridge film of 1940 is close to, but not exactly, textbook resonance — the wind was not a rhythmic driver; the bridge's own twisting fed energy back on itself (aeroelastic flutter, a self-excited oscillation). "Resonance-like disaster, driven by self-excitation" earns the mark and the examiner's respect.

Limiting cases

case	result	meaning
$b \to 0$ , free	amplitude constant forever	the idealised SHM of earlier modules
light damping	$f' \lesssim f_0$ , geometric fade	pitch survives, loudness dies — the guitar string
critical damping	fastest no-overshoot return	design point: suspensions, door closers, meter needles
overdamped	slow ooze home	more damping now means slower settling
driven, $f_d \ll f_0$	$A \approx$ driver's stroke, in step	quasi-static following
driven, $f_d \gg f_0$	$A \to 0$	inertia wins; the mass stands still
driven, $f_d = f_0$ , $b \to 0$	$A$ grows toward destruction	resonance with no ceiling — the catastrophe limit

Common trap“The driven oscillator vibrates at its natural frequency”

It does not — and this single confusion costs more marks than everything else in the module combined. In steady state a forced oscillator vibrates at the driver's frequency $f_d$ , full stop; the natural frequency $f_0$ only decides the amplitude of that response (huge when $f_d \approx f_0$ , tiny far away). Where does $f_0$ show itself? Only in free oscillation — pluck it, release it, and the transient rings at (essentially) $f_0$ while it lasts. The companion misstatements to refuse on sight: "at resonance the amplitude becomes infinite" (damping always caps it — infinite only in the fictional $b = 0$ limit), and "damping changes the pitch strongly" (light damping's frequency shift is second-order small; the guitar string fades without detuning).

Worked examples

1 · Board level — the decay bookkeeping (CBSE/ISC/A-Level pattern)

The amplitude of a lightly damped oscillator falls to half its initial value in 20 complete cycles. (a) What fraction of the mechanical energy remains? (b) What will the amplitude be after 40 cycles? (c) Does the period change noticeably during the decay?

Step 1. Energy tracks amplitude squared: $\dfrac{E}{E_0} = \left(\dfrac{A}{A_0}\right)^2 = \left(\dfrac12\right)^2 = \boxed{\dfrac14}$ — half the amplitude, a quarter of the energy; three-quarters has already gone to heat and stirred air.

Step 2. Equal times, equal fractions: the next 20 cycles repeat the same ratio. After 40 cycles $A = A_0 \times \tfrac12 \times \tfrac12 = \boxed{A_0/4}$ (and $E = E_0/16$ ). Never subtract — decay multiplies.

Step 3. No. Light damping leaves the frequency essentially at $f_0$ (the shift is second-order); the fade is in amplitude alone. This is why a struck tuning fork stays on pitch while it dies — and it is the sentence the examiner wants verbatim.

2 · Competitive level — half-life of a damped block (JEE-Main pattern, ~2 minutes)

A $0.50\ \mathrm{kg}$ block oscillates on a spring with damping constant $b = 0.10\ \mathrm{kg/s}$ . Using $A(t) = A_0e^{-bt/2m}$ , find (a) the time for the amplitude to halve, (b) the time for the energy to halve. (Take $\ln 2 = 0.693$ .)

Step 1. Amplitude halves when $e^{-bt/2m} = \tfrac12$ , i.e. $\dfrac{bt}{2m} = \ln 2$ : $t_{A/2} = \dfrac{2m\ln 2}{b} = \dfrac{2 \times 0.50 \times 0.693}{0.10} = \boxed{6.9\ \mathrm{s}}$ .

Step 2. Energy halves when $e^{-bt/m} = \tfrac12$ — the exponent is twice as big, so the time is half: $t_{E/2} = \dfrac{m\ln 2}{b} = \boxed{3.5\ \mathrm{s}}$ .

Step 3. Sense audit: energy must reach its half-life first (it decays twice as fast); $3.5 < 6.9$ ✓. When the amplitude finally halves at 6.9 s, the energy is already down to a quarter ✓. Timing note: ~2 min. Pattern-match, don't derive: the exponent doubles between $A$ and $E$ , so every energy time is half the corresponding amplitude time. Distractor watch: $t_{E/2} = 13.9$ s (doubling instead of halving) sits among the options every time this is asked.

3 · Competitive level — pushing the swing (NEET/JEE-Main pattern, ~2 minutes)

A child's swing has ropes of effective length $2.5\ \mathrm{m}$ . (a) At what frequency should a parent push, one shove per cycle, for maximum amplitude? (b) The parent instead pushes every $2.0\ \mathrm{s}$ . Qualitatively, what happens? ( $g = 9.8\ \mathrm{m/s^2}$ .)

Step 1. The swing is a pendulum: $T_0 = 2\pi\sqrt{L/g} = 2\pi\sqrt{2.5/9.8} = 2\pi \times 0.505 = 3.17\ \mathrm{s}$ , so the resonant push rate is $f_0 = 1/T_0 = \boxed{0.32\ \mathrm{Hz}}$ — one push roughly every three seconds.

Step 2. Pushing every 2.0 s means $f_d = 0.50$ Hz — well above $f_0$ . The swing settles into a small steady oscillation at the pushing rhythm (0.50 Hz — the driver always wins the frequency), with many shoves landing against the motion. Result: a modest, jerky response, far below the resonant amplitude.

Step 3. The design lesson: to build amplitude you match the natural rhythm and let each deposit arrive in phase — which is also why "well-meaning but mistimed pushing" is the standard exam metaphor for off-resonance driving. Timing note: ~2 min. Part (a) is the pendulum period in costume; part (b) is the two-sentence catechism — frequency: driver's; amplitude: small off-resonance. Saying the swing "oscillates at 0.32 Hz anyway" is the planted error.

Exam traps

Steady state = driver's frequency. Not the natural frequency, not an average of the two. The natural frequency governs only the size of the response (and the brief transient at switch-on).

Amplitude decay is multiplicative. "Falls by 4 cm every cycle" is the wrong model; "falls to 90% every cycle" is right. After $n$ cycles, multiply the ratio $n$ times. Linear-subtraction answers are always among the MCQ options.

Energy runs at double speed. $E \propto A^2$ : amplitude ratio $r$ per cycle means energy ratio $r^2$ ; amplitude half-life $\Leftrightarrow$ energy quarter-point; energy half-life is half the amplitude half-life.

Resonance amplitude is finite. Damping is the ceiling: the peak height is set by $b$ , and only the fictional undamped oscillator diverges. "Infinite amplitude at resonance" is an automatic zero in a reasoning question.

Critical damping is the optimum, not the maximum. Fastest no-overshoot settling happens at critical; overdamping settles more slowly. Car-suspension and galvanometer questions test exactly this reversal.

Damping's effect on frequency is slight. Lightly damped oscillators keep (essentially) their pitch; only near-critical damping drags the frequency visibly down, and past critical there is no frequency at all.

Scope notes by board. CBSE/NEET treat this module qualitatively (curves, comparisons, the swing story) — recent NCERT editions trimmed the formulas, so the decay mathematics above is your safety margin, not a requirement. JEE-Advanced names damped and forced oscillation with resonance explicitly — Example 2's formula work is in scope there. CIE/IB want the curves, critical-damping applications and Barton/bridge stories.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Cambridge and Terling Place, Essex

John William Strutt, third Baron Rayleigh, wrote the book on this module — literally. His Theory of Sound (1877) treats damped and driven oscillations systematically for the first time: the exponential envelope, the steady state at the driver's frequency, the resonance response and its damping-controlled sharpness are all set out in the notation physics still uses. Rayleigh was famously double-tracked: a meticulous experimentalist (he isolated argon, earning the 1904 Nobel Prize) and a master of applied mathematics who could extract a resonance curve from a page of algebra. His name survives across physics — Rayleigh scattering (why the sky is blue), Rayleigh waves (how earthquakes shake buildings — into resonance, when they are unlucky), the Rayleigh criterion in optics — but his deepest legacy is the habit this module teaches: treat every vibrating thing, from a wine glass to a bridge deck, as one damped-driven oscillator with one response curve.

History

1831

— The Broughton suspension bridge near Manchester collapses under soldiers marching in step; armies worldwide adopt "break step on bridges."

1850

— Stokes analyses fluid drag proportional to velocity — the

-bv

law that models light damping.

1877

— Rayleigh's Theory of Sound gives damped and forced oscillations their modern mathematical form.

1940

— The Tacoma Narrows bridge tears itself apart in a 68 km/h wind — self-excited flutter, the resonance family's most famous film.

2000

— London's Millennium Bridge closes after two days as pedestrian footfalls lock into its lateral sway; 91 dampers later, it reopens.

Going further

Texts. NCERT Physics Class 11, Ch. 14 (damped and forced oscillations sections, where retained) · HC Verma, Concepts of Physics Vol. 1, Ch. 12 · Resnick–Halliday–Walker Ch. 15 · The UG deep dives: The Damped Harmonic Oscillator (three regimes, $Q$ ), Driven Oscillations and Resonance (the full response curve and phase story).

Provenance. All results are standard. A verifier should recompute: the derivation block's $0.9^{10} = 0.35$ , $\sqrt{0.35} = 0.59$ ; Example 1's ¼ and $A_0/4$ ; Example 2's $t_{A/2} = 6.9$ s and $t_{E/2} = 3.5$ s from $2m\ln2/b$ and $m\ln2/b$ ; Example 3's $T_0 = 3.17$ s and $f_0 = 0.32$ Hz.

Prerequisites. The Spring–Block Oscillator · Energy in SHM · The Simple Pendulum. Leads to. The Damped Harmonic Oscillator (UG deep dive) · Resonance (UG deep dive) · Waves & Sound (coming track).

Why does light damping leave the frequency almost unchanged while it destroys the amplitude?

Because the two live at different orders of the damping strength. The energy drain per cycle is first order in $b$ — directly proportional — so amplitude feels damping immediately. The frequency correction is second order: $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$ , and squaring a small number makes it tiny. A system losing a full 5% of its energy per cycle has its frequency lowered by about 0.001%. That hierarchy is a gift to civilisation: it means clocks and musical instruments can afford friction — they get quieter, not slower.

What's the actual difference between free, damped, forced and maintained oscillations? The words blur together.

Free: displace, release, no further interference — the system rings at its natural frequency, forever in the ideal case. Damped: free plus friction — same natural rhythm (essentially), amplitude fading geometrically. Forced (driven): a periodic external force acts throughout — after transients, the system oscillates at the driver's frequency with constant amplitude. Maintained (sustained): a device tops up exactly the energy damping removes each cycle at the system's own rhythm — a pendulum clock's escapement, a child pumping her own swing — so the system runs at its natural frequency with constant amplitude. The last two differ in who commands the frequency: an external driver imposes its own; a maintaining mechanism obeys the oscillator's.

Why does the amplitude at resonance not grow to infinity?

Follow the budget. The driver deposits a roughly fixed amount of energy per cycle (force × distance moved in phase). Damping withdraws an amount proportional to the square of the amplitude (the derivation block's Step 1). As $A$ climbs, withdrawals grow quadratically while deposits grow only linearly with $A$ — so there is always an amplitude at which the two meet, and there the system settles. Lighter damping moves that meeting point higher (taller peak); zero damping never meets it, which is the fictional divergence. Real catastrophes happen not at infinity but when the finite resonant amplitude exceeds what the structure can take.

Was the Tacoma Narrows collapse resonance or not? Different sources say different things.

It sits one door down the same corridor. Textbook resonance needs a periodic external driver whose frequency matches a natural mode. At Tacoma the wind was steady, not rhythmic — the periodicity was manufactured by the bridge itself: as the deck twisted, it shed air vortices in time with its own twisting, so each oscillation invited a stronger push. That feedback loop is called aeroelastic flutter, a self-excited oscillation. The energy-stacking endgame looks exactly like resonance (which is why the film lives in this chapter), but the driver was internal. An exam answer that says "resonance-like self-excited flutter, not simple forced resonance" is the gold-standard version; plain "resonance" still earns the point at board level.

Does a microwave oven work by hitting the resonant frequency of water molecules?

No — it is the most popular resonance myth in physics classrooms. Water's molecular resonances sit at terahertz frequencies, thousands of times above the oven's 2.45 GHz. Microwaves heat by dielectric loss: the field flips polarity billions of times a second, polar water molecules keep trying to realign, and their rotational jostling against neighbours dissipates energy as heat — friction, not resonance. The tell: a true resonance would be absurdly sharp, and food's absorption is broad. If an exam or interviewer floats the myth, the phrase to deploy is "forced rotation with dielectric loss, well below any resonance."

What is the "quality factor" I see on graphs and in JEE solutions?

A single number rating how lightly damped an oscillator is: $Q$ counts (up to a factor of $2\pi$ ) how many cycles the system can ring before damping eats most of its energy — equivalently, how tall and narrow its resonance peak is. A car suspension has $Q \sim 1$ (settle immediately, never ring), a guitar string $Q \sim 10^3$ , a quartz watch crystal $Q \sim 10^5$ , LIGO mirror suspensions $Q \sim 10^9$ . At HS you need recognition, not machinery: high $Q$ = slow decay = sharp resonance; low $Q$ = fast decay = broad mound. The full treatment is the UG damped-oscillator deep dive.

Why do soldiers break step on bridges but crowds don't seem to cause problems?

Soldiers are a coherent driver: hundreds of feet landing at the same 2 Hz put their entire force into one frequency, and if a bridge mode lives nearby, deposits stack every cycle — the 1831 Broughton collapse was exactly this. A random crowd smears its footfalls across all phases; contributions mostly cancel, like pushing a swing at random moments. The Millennium Bridge found the loophole: its lateral sway synchronised the crowd — people widen their stance and push sideways in time with a swaying deck to keep balance — turning random pedestrians into an accidental marching column. The fix was not stiffening but damping: 91 dampers to flatten the peak.