Which formulas does this calculator use (mechanical)?

For a mass–spring–damper with mass m, stiffness k, and viscous damping c: ωₙ = √(k/m), fₙ = ωₙ/(2π), critical damping c_c = 2√(km), damping ratio ζ = c/c_c, Q = 1/(2ζ), damping factor α = c/(2m), and damped natural frequency ω_d = ωₙ√(1 − ζ²) for ζ<1.

Which formulas does this calculator use (RLC)?

For ideal L and C: ω₀ = 1/√(LC), f₀ = ω₀/(2π). Series RLC: α = R/(2L), ζ = α/ω₀ = (R/2)√(C/L), Q = (1/R)√(L/C). Parallel RLC: α = 1/(2RC), ζ = α/ω₀ = (1/2R)√(L/C), Q = R√(C/L). Bandwidth Δf = f₀/Q (small damping).

Does it run privately?

Yes. All calculations run in your browser. Only your analytics script loads; no inputs leave the page.

SI: m in kilograms (kg), k in N/m, c in N·s/m. For RLC: L in henries (H), C in farads (F), R in ohms (Ω). Results are in rad/s, Hz, seconds, and ohms as appropriate.

Resonance & Natural Frequency Calculator

Choose Mass–Spring–Damper or RLC (Series/Parallel). Get ωₙ, fₙ, ζ, Q, α, ω_d, and bandwidth. Clean units, deep links, and a simple diagram—engineer-friendly and physics-curious.

Inputs

System Switch between mechanical and electrical models.

Mass m (kg) ≥ 0; use kg.

Stiffness k (N/m) Spring constant; must be ≥ 0.

Damping c (N·s/m) Viscous damping; 0 for undamped.

Precision (digits)

Tip: Press Ctrl/Cmd + Enter to calculate. The URL updates so you can bookmark or share your inputs.

Results

Resonance 101: Mechanical ↔ Electrical

Resonance is a universal phenomenon that links physics, engineering, and everyday life. A playground swing, a tuning fork, and a tuned radio all rely on the same mathematics of oscillation. Whenever energy can slosh back and forth between two forms—kinetic vs. potential, electric vs. magnetic—the system has a natural frequency at which it prefers to vibrate. At that frequency, even small inputs can build to large responses. Understanding resonance is essential for designing safe mechanical systems, efficient circuits, and everything from skyscrapers to smartphone antennas.

In mechanics, the simplest model is the mass–spring–damper. With mass m, stiffness k, and viscous damping c, the natural angular frequency is ωₙ = √(k/m), corresponding to fₙ = ωₙ/(2π). The damping ratio ζ = c / (2√(km)) determines whether the system is underdamped (oscillatory), critically damped (fastest non-oscillatory return), or overdamped (slow return). The quality factor Q = 1/(2ζ) measures sharpness: high-Q systems ring for many cycles, low-Q systems die out quickly. Engineers also track the decay constant α = c/(2m) and the damped natural frequency ω_d = ωₙ√(1 − ζ²), which is slightly less than the undamped value.

Electrical engineers see the same formulas in RLC circuits. With inductance L, capacitance C, and resistance R, the undamped resonance is ω₀ = 1/√(LC), f₀ = ω₀/(2π). In a series RLC, damping arises from R/(2L), giving ζ = (R/2)√(C/L) and Q = (1/R)√(L/C). In a parallel RLC, the roles invert: ζ = (1/2R)√(L/C) and Q = R√(C/L). The bandwidth is approximately Δf = f₀ / Q, a key spec for filters, amplifiers, and radio receivers.

Units and sanity checks: Use SI: m in kg, k in N/m, c in N·s/m, L in henries, C in farads, R in ohms. Increasing stiffness k or reducing mass m raises frequency. In circuits, increasing resistance R lowers Q for a series RLC but raises Q for a parallel RLC. These cross-checks help spot unit mistakes. In both domains, the resonance story is the same: energy storage, dissipation, and feedback combine to create rich dynamics that can be harnessed—or avoided.

5 Fun Facts about Resonance & Natural Frequency

Bridge swing lesson

The Tacoma Narrows Bridge collapse in 1940 wasn’t from high winds alone—it was wind energy locking onto the bridge’s torsional natural frequency, feeding the oscillation every cycle.

Bridge lore

Skyscraper counterweights

Super-tall buildings hide hundreds of tons of tuned mass dampers—giant pendulums or water tanks that are deliberately tuned to the tower’s sway frequency to cancel motion.

City calm

Quartz keeps time

The tiny tuning-fork crystal inside a quartz watch resonates at 32,768 Hz with a Q in the tens of thousands, so it can count seconds by simply dividing a super-stable tone.

Clockwork crystal

Radio scavenger hunt

An RLC circuit tuned to a station is literally a frequency-selective “net.” Only currents near its natural frequency build up, letting your receiver ignore everything else on the dial.

Signal selectivity

Nanoscopic feelers

Atomic force microscopes vibrate a cantilever at resonance and watch its frequency shift by a few Hz when it feels atomic forces—basically using resonance as a force magnifier.

Nano detective