This simulation models a mass-spring-damper oscillator driven by a periodic external force F(t) = F0·cos(omega·t). When the driving frequency omega approaches the system's natural frequency omega0 = sqrt(k/m), the steady-state amplitude grows dramatically — this is mechanical resonance. Users can adjust mass, spring stiffness, damping, and driving frequency in real time and observe how the resonance curve A(omega) and phase-lag curve phi(omega) change.
Driven resonance is central to physics and engineering: radio tuners exploit it to select broadcast frequencies, musical instruments rely on it to amplify specific pitches, and structural engineers must design bridges and buildings to avoid destructive resonance with wind or earthquake forcing frequencies.
Driven resonance occurs when an oscillating system is subjected to a periodic external force whose frequency matches the system's natural frequency. At this condition the system absorbs energy most efficiently from the driving force, causing the amplitude of oscillation to reach a sharp maximum. The natural frequency is determined by the physical properties of the system — for a spring-mass oscillator it equals sqrt(k/m).
Set the mass m, spring constant k, and damping b using the left panel sliders, then slowly drag the driving frequency omega slider toward the displayed natural frequency omega0. Watch the displacement amplitude spike sharply as omega approaches omega0. For the clearest effect, reduce damping b to a small value (below 1 N·s/m) and use the "Auto-sweep omega" button to scan automatically from low to high frequency while the resonance curve is plotted live.
Below the natural frequency the oscillator moves nearly in phase with the driving force (phase lag close to 0 degrees). At exact resonance the phase lag is always exactly 90 degrees, regardless of how much damping is present. Above resonance the phase lag approaches 180 degrees, meaning the mass moves opposite to the applied force. This 90-degree signature at resonance is the basis for many frequency-detection techniques.
The quality factor Q = sqrt(m·k) / b measures how sharply peaked the resonance is. A high-Q system (low damping) has a narrow, tall resonance peak and rings for many cycles before decaying. A low-Q system (high damping) has a broad, flat peak. Q also equals the ratio of the energy stored in the oscillator to the energy dissipated per radian of oscillation, and it approximates the number of oscillation cycles needed for the amplitude to fall to 1/e of its starting value after the driving force is removed.
The Tacoma Narrows Bridge collapse in November 1940 is the most cited example, though the mechanism was aeroelastic flutter rather than pure resonance — wind created a self-reinforcing torsional oscillation. Purer resonance failures include the Angers Bridge (1850), where marching soldiers' footsteps matched the bridge's natural frequency and caused it to collapse, prompting the military rule of breaking step on bridges. Resonance is also responsible for the shattering of wine glasses by a sustained pure tone at the glass's natural frequency.
A widespread misconception is that an undamped system at resonance reaches infinite amplitude instantly. In reality, amplitude grows without bound only asymptotically over time in an idealized undamped system, and in practice all real systems have some damping that limits the peak. Another misconception is that the amplitude peak occurs exactly at omega0; for a damped oscillator the amplitude peak is slightly below omega0 at omega_peak = sqrt(omega0^2 - b^2/(2m^2)), shifting lower as damping increases.
The mathematical treatment of forced harmonic oscillations was developed in the 18th and 19th centuries. Daniel Bernoulli studied vibrating strings and superposition around 1750. Jean le Rond d'Alembert and Leonhard Euler contributed foundational wave and oscillation equations. The complete linear damped-driven oscillator model, including the resonance amplitude formula, emerged from the synthesis of Newtonian mechanics with the work of scientists including George Gabriel Stokes (viscous damping, 1851) and later Hermann von Helmholtz, who connected resonance to acoustics and hearing in his 1863 work "On the Sensations of Tone."
Directly related phenomena include simple harmonic motion (undriven spring-mass), coupled oscillators and normal modes, parametric resonance (where a system parameter varies periodically), and nonlinear oscillators such as the Duffing oscillator. Resonance also appears in wave physics as standing waves and in quantum mechanics as Rabi oscillations. The cloth and fluid simulations on this site involve oscillatory instabilities that share mathematical structure with driven resonance.
Engineers exploit driven resonance in radio and television tuners (an LC circuit resonates at the selected broadcast frequency), quartz crystal oscillators (used in clocks and processors, Q factors above 10,000), magnetic resonance imaging (MRI protons are driven at their Larmor resonance frequency in a magnetic field), and MEMS sensors (microscale resonators detect acceleration, pressure, or mass changes via frequency shifts). Anti-resonance design is equally important: suspension systems, vibration isolators, and tuned mass dampers are engineered to prevent structures from resonating destructively.
Active research areas include nonlinear resonance in micro- and nano-electromechanical systems (NEMS), where quantum effects become relevant at cryogenic temperatures; topological phononic crystals that guide mechanical waves using resonance band gaps; stochastic resonance, in which adding noise to a noisy system counterintuitively improves signal detection; and vibrational energy harvesting, where ambient mechanical vibrations are converted to electrical power via piezoelectric resonators tuned to the environment's dominant frequency. Parametric amplification using driven resonance is also being explored for quantum computing qubit readout.