Mechanical Resonance Frequency Tool
Calculate natural and damped resonance frequencies for spring-mass systems, cantilever beams, and simply-supported beams. Includes damping ratio, quality factor, peak magnification, and a live frequency-response plot of the magnification curve.
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About Mechanical Resonance
Every elastic structure — a guitar string, a car suspension, a steel bridge, a skyscraper — has at least one natural frequency at which it prefers to vibrate. When driven by an external force at that frequency, vibration amplitude can grow dramatically — that's resonance. This tool calculates resonance for three classic system families that cover the majority of textbook and engineering cases.
Spring-mass system (single degree of freedom)
The simplest oscillator: a mass m attached to a spring of stiffness k. The undamped natural angular frequency is ω₀ = √(k/m), and the frequency in Hz is f₀ = ω₀ / (2π). A car suspension typically has k ≈ 30 kN/m and m ≈ 350 kg per corner, giving f₀ ≈ 1.5 Hz — comfortably below the 4–8 Hz range where human bodies become sensitive to vertical motion. A typical hand-wound spring with k = 100 N/m holding a 0.5 kg mass resonates at f₀ ≈ 2.25 Hz.
Cantilever beam (one end fixed, one end free)
The classic "diving board" geometry. Resonance frequencies are f_n = (β_n·L)² · √(EI / (ρA·L⁴)) / (2π), where β_n·L takes specific values (1.8751, 4.6941, 7.8548 for the first three modes). The fundamental is roughly 0.56 · √(EI / (m·L³)) where m is the total beam mass. Note that the modes are not simple integer multiples — f₂ ≈ 6.27·f₁, f₃ ≈ 17.55·f₁. This is why a vibrating ruler held at the edge of a desk doesn't sound like a pitched musical instrument.
Simply-supported beam (both ends pinned)
A bridge span or piano string approximation. Here β_n·L = n·π exactly, so f_n = (n·π)² · √(EI / (ρA·L⁴)) / (2π) = n²/2 · √(EI / (μL⁴)) where μ = ρA. The mode ratios are square integers: f₂ = 4·f₁, f₃ = 9·f₁, f₄ = 16·f₁. A piano string's harmonics are close to (but slightly above) these ratios — the small discrepancy is called inharmonicity and is responsible for the characteristic "stretched" tuning of a piano.
Damping ratio ζ — how peaky is the resonance?
Damping ratio ζ (zeta) controls how sharply the system responds at resonance. ζ = 0: no damping, the magnification at resonance is infinite (the system would amplify forever — Tacoma Narrows territory). ζ < 1: underdamped, oscillates with decreasing amplitude after a transient. ζ = 1: critically damped, returns to rest as fast as possible without overshooting. ζ > 1: overdamped, sluggish return without oscillation. Practical values: a tuning fork ζ ≈ 0.0001 (Q ≈ 5000), a car suspension ζ ≈ 0.3 (Q ≈ 1.7), a damped door closer ζ ≈ 1.
Frequently Asked Questions
What's the difference between f₀ and f_d?
What is Q-factor and how does it relate to ζ?
Q = 1/(2ζ). It also equals the peak magnification at resonance for lightly damped systems. A high-Q system (Q > 100) has a sharp narrow resonance peak; a low-Q system (Q < 5) has a broad shallow peak. Quartz crystal oscillators have Q ≈ 10,000–100,000 — extraordinarily sharp resonance. Car suspensions deliberately have Q ≈ 1.5 to avoid bouncy ride.Why don't cantilever beam modes follow integer ratios?
cos(βL)·cosh(βL) = −1. This transcendental equation has roots at βL = 1.875, 4.694, 7.855, … — not at integer multiples. Frequency scales as (βL)², so f₂/f₁ ≈ 6.27, f₃/f₁ ≈ 17.55. That's why a tuning fork's overtones aren't harmonic — fork prongs are cantilevers, so the second mode is roughly 6× the fundamental, not 2× like a string.Why does increasing damping reduce the resonance peak?
What is the "frequency response curve" showing?
M(r) = 1/√((1−r²)² + (2ζr)²) against the frequency ratio r = ω/ω₀. M is the steady-state amplitude ratio — how much the system amplifies an input vibration at frequency ω relative to the same force applied statically (r=0). At r=1 (resonance) and low ζ, the peak rises sharply. At very high frequencies (r → ∞), M → 0 — the system can't keep up with fast driving. The dashed line at M=1 marks "unity response" — input passes through unchanged.