Think of a kid on a playground swing. Give little pushes at just the right moments — once each time the swing comes back to you — and it climbs higher and higher with hardly any effort. Push at the wrong moments and you fight the swing as often as you help it, so it barely moves. That single everyday fact is resonance: a wiggle delivered at the system's favourite rhythm builds up a huge response, while the same wiggle at the wrong rhythm does almost nothing.
Every springy thing has a favourite rhythm, its natural frequency $\omega_0$. For a mass $m$ on a spring of stiffness $k$ it is set entirely by those two numbers: a stiffer spring or a lighter mass swings faster. Real systems also lose energy to friction or drag — we call that damping, measured by the rate $\gamma$ — so a swing left alone slowly dies down. The third player is the driving frequency $\omega$: how fast you push.
Here is the heart of it. The steady response is biggest when you drive close to $\omega_0$, and how big it gets depends on the damping. With the sim's defaults ($k=10\text{ N/m}$, $m=1\text{ kg}$) the favourite rhythm is $\omega_0=\sqrt{10}\approx 3.16$ rad/s. A steady $F_0=1\text{ N}$ push applied slowly would only stretch the spring by about $F_0/k = 0.1$ m. But push right at $\omega_0$ and the amplitude swells to roughly $A_{\max}\approx QF_0/k$, where the quality factor $Q=\omega_0/2\gamma$ counts how lightly damped the system is. With $Q\approx 6$ that same 1 N push now produces about $0.6$ m — six times larger, for free.
The precise amplitude-versus-frequency law shows exactly where the peak sits and why heavy damping flattens it:
The sliders map straight onto these symbols: m and k set $\omega_0$, b sets the damping $\gamma$ (and so $Q$), and F₀, ω are the strength and rhythm of your push. When $\gamma$ grows past $\omega_0/\sqrt{2}$ the square root above turns imaginary — the peak vanishes entirely, which is why a heavily damped door-closer never overshoots.
Try this in the sim above. (1) In Driven (Steady) mode, slowly drag ω up through 3.16 rad/s and watch the amplitude readout spike as you cross $\omega_0$. (2) Now raise the damping b and repeat — the peak gets shorter and broader, and $Q$ drops. (3) Switch to Frequency Sweep and watch the whole $A(\omega)$ resonance curve draw itself, with the green $\omega_0$ marker showing where the hump lives.
Section 03
Equations & Derivation
Equation of Motion (Driven Damped Oscillator)
$$m\ddot x + b\dot x + kx = F_0 \cos(\omega t)$$
Standard Form
$$\ddot x + 2\gamma\dot x + \omega_0^2\,x = \frac{F_0}{m}\cos(\omega t),\qquad \omega_0 = \sqrt{\frac{k}{m}},\quad \gamma = \frac{b}{2m}$$
$$Q = \frac{\omega_0}{2\gamma} = \frac{\omega_0 m}{b} = 2\pi \times \frac{\text{Energy stored}}{\text{Energy lost per cycle}}$$
Symbol Definitions
Symbol
Meaning
SI Unit
$m$
Mass
kg
$k$
Spring constant
N m⁻¹
$b$
Damping coefficient (linear drag)
kg s⁻¹
$\omega_0$
Natural (undamped) frequency $\sqrt{k/m}$
rad s⁻¹
$\gamma$
Damping rate $b/2m$
s⁻¹
$\omega_d$
Damped frequency $\sqrt{\omega_0^2 - \gamma^2}$
rad s⁻¹
$\omega$
Driving frequency
rad s⁻¹
$Q$
Quality factor — sharpness of resonance
dimensionless
$F_0$
Drive force amplitude
N
Step 1Newton's second law. Spring force $-kx$, drag force $-b\dot x$, drive force $F_0\cos\omega t$. Sum $= m\ddot x$. Divide by $m$ to get standard form with $\gamma = b/2m$ and $\omega_0^2 = k/m$.
Step 2Free oscillation ($F_0 = 0$). Try $x = e^{rt}$. Characteristic equation $r^2 + 2\gamma r + \omega_0^2 = 0$ gives $r = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$. Three cases by sign of discriminant: underdamped (oscillates with envelope $e^{-\gamma t}$), critically damped (returns fastest, no overshoot), overdamped (slow non-oscillatory return).
Step 3Driven steady state. Try $x_{\text{ss}} = A\cos(\omega t - \phi)$. Substitute and equate $\cos$ and $\sin$ coefficients. After algebra, get the boxed amplitude and phase formulas.
Step 4Resonance. Maximise $A(\omega)$ by setting $dA/d\omega = 0$. The amplitude peaks at $\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$ — slightly below $\omega_0$ (and below $\omega_d$). For light damping $\gamma \ll \omega_0$: $\omega_{\text{res}} \approx \omega_0$, and $A_{\max} \approx F_0/(2m\gamma\omega_0) = QF_0/k$.
Step 5Energy interpretation of Q. Stored energy $\propto A^2$. Per cycle $\Delta E/E \approx 4\pi\gamma/\omega_0 = 2\pi/Q$. So $Q = 2\pi \times$ (energy stored)/(energy lost per cycle). Tuning fork: $Q \sim 10^3$. Quartz crystal: $Q \sim 10^5$. Atomic clock: $Q \sim 10^{11}$.
How simulation variables map to the equations
Sliders m, k, b set the natural frequency $\omega_0 = \sqrt{k/m}$ and damping $\gamma = b/2m$. Drive sliders F₀ and ω control the external sinusoidal force. Set F₀ = 0 and adjust b to see underdamped/critical/overdamped regimes. Set F₀ > 0 and sweep ω through $\omega_0$ to see resonance build up. The "Frequency Sweep" mode automates this and plots the resonance curve $A(\omega)$.
Reference: A. P. French — Vibrations and Waves (MIT Introductory Physics Series), Ch. 3 §3-3 to §3-7 and Ch. 4: "The Forced Vibration of a Damped Oscillator"; Halliday, Resnick & Walker — Fundamentals of Physics, 10th Ed., §15-8 to §15-9.
Section 04
Frequently Asked Questions
Car suspension (designed near critical damping), playground swings (you push at the natural frequency = driving), microwave ovens (water dipole resonance at 2.45 GHz), MRI machines (proton spin precession at Larmor frequency), every musical instrument (string + air resonator), the Tacoma Narrows Bridge collapse (1940 — wind-driven aerodynamic flutter at the bridge's natural frequency).
💡 Resonance is everywhere — it's how energy gets selectively transferred.
A mass on a spring with adjustable damping and an optional sinusoidal driving force. Toggle modes to compare free oscillation, damped decay (three regimes), and steady-state driven motion. Switch to "Frequency Sweep" to watch the resonance curve $A(\omega)$ build up as the driver scans through frequencies. The graph tabs show $x(t)$, $v(t)$, energy decay, and the steady-state response curves.
💡 Watch the "Frequency Sweep" mode — that peak is resonance.
Underdamped systems oscillate around equilibrium — they return quickly initially but overshoot multiple times. Overdamped systems return monotonically but slowly because the spring force is mostly cancelled by drag. Critical damping ($\gamma = \omega_0$) sits exactly at the boundary: monotone return at the maximum possible rate. That's why it's engineered into car suspensions, automatic door closers, and analogue instrument needles.
💡 Critical damping = no overshoot + fastest possible return.
When the driver pushes one way and the mass responds the same way, $\phi = 0$ (low frequency). When the driver pushes and the mass moves opposite, $\phi = \pi$ (high frequency). At resonance ($\omega = \omega_0$ for light damping), $\phi = \pi/2$ exactly — the velocity is in phase with the drive, maximising power transfer to the oscillator.
💡 At resonance, drive is in phase with velocity — maximum power transfer.
For very weak damping the amplitude peaks essentially at $\omega_0$. As $\gamma$ increases, the peak shifts to $\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$ — strictly below $\omega_0$. When $\gamma > \omega_0/\sqrt{2}$, the formula breaks down: there is no peak at all, just a monotonically decreasing $A(\omega)$. Heavily damped systems do not resonate.
💡 Heavy damping kills the resonance peak entirely.
Not classical resonance — modern engineers call it aerodynamic flutter, a self-excited instability where the wind-induced oscillation creates lifting forces that reinforce the next cycle. It's a related but distinct mechanism. True resonance failures do happen though: the Broughton Suspension Bridge collapsed in 1831 when soldiers marched in step at its natural frequency, leading to the rule that troops "break step" on bridges.
💡 Tacoma was flutter; Broughton was true resonance — both deadly.
If you push out of phase with the swing's motion, you sometimes accelerate it and sometimes decelerate it — net energy transfer averages near zero. Push at the resonance frequency and exactly when the swing is at maximum velocity in your push direction, every push adds energy to the system, building amplitude rapidly. This is the kid's intuition behind "pumping" a swing.
💡 Energy is transferred maximally only at resonance, in phase with velocity.
Resource: MIT OCW 8.03 — Vibrations and Waves, Lectures 1–4 (Walter Lewin); HyperPhysics — Driven Harmonic Oscillator; Khan Academy — Resonance.
Section 05
Common Misconceptions
❌ At resonance, the response is infinite.
✅ Only an undamped, ideal driven oscillator has unbounded amplitude at $\omega = \omega_0$. Any real damping limits the peak to $A_{\max} = QF_0/k$ — finite, but possibly destructively large. Resonance amplification is bounded by $Q$, not by infinity. Tacoma Narrows did not have infinite amplitude — it had amplitudes large enough to cause material failure.
📖 HRW 10th Ed., §15-9, Eq. 15-43.
❌ Resonance occurs exactly at $\omega = \omega_0$.
✅ The amplitude resonance is at $\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$, which is below $\omega_0$. Velocity resonance is exactly at $\omega_0$. Energy/power absorption resonance is also at $\omega_0$. So which "resonance" you mean depends on what you're measuring — amplitude or power. Most textbooks blur this distinction.
📖 A. P. French — Vibrations and Waves, MIT, §4-3.
❌ Critical damping means "perfectly tuned" damping — the optimal value.
✅ "Optimal" depends on application. For shock absorbers, a value slightly below critical (around $0.6$–$0.7$ of critical) gives a more comfortable ride at the cost of slight overshoot. For ammeters and analytical balances, exactly critical is preferred. There is no universal "best" damping — it's an engineering trade-off.
📖 Marion & Thornton — Classical Dynamics, 5th Ed., §3.5.
❌ A resonant system stores up energy indefinitely.
✅ In steady state, the driver's power input exactly balances the damper's dissipation rate. No energy accumulates beyond the steady-state amount. Removing the driver causes the energy to decay at rate $2\gamma$. The "infinite buildup" picture only applies in the unphysical undamped limit.
📖 French — Vibrations and Waves, MIT, §4-5.
❌ Free oscillation always occurs at $\omega_0$.
✅ Free oscillation occurs at $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$, which is below the natural frequency $\omega_0$. A heavily damped pendulum oscillates more slowly than a frictionless one. In the limit of critical damping ($\gamma = \omega_0$), $\omega_d = 0$ — the system stops oscillating altogether.
📖 HRW 10th Ed., §15-8, Eq. 15-42.
❌ Q-factor is just a measure of how long the oscillation lasts.
✅ Q has multiple equivalent definitions: (1) ratio of resonance frequency to bandwidth at half-power; (2) $2\pi$ times stored energy over energy lost per cycle; (3) number of radians the system oscillates before energy decays by factor $e$. All three give the same number — but Q is more about spectral selectivity than time. A radio receiver needs high Q to separate stations; a shock absorber wants low Q.
📖 French — Vibrations and Waves, MIT, §3-5.
Misconception research: Greenslade Jr. (1991), "The American Journal of Physics" 59, 829 — student understanding of resonance; Sokoloff & Thornton (1997), "Real-time physics" Active Learning Vol. 3.