Mechanical Resonance Frequency Tool

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 phenomenon is also called a structural resonance or eigenfrequency in finite-element analysis, and the associated deformation shape is the mode shape or eigenvector. This mechanical resonance calculator covers three classic system families — spring-mass, cantilever beam, and simply-supported beam — that account for the majority of textbook and engineering cases. For room and building resonance driven by mechanical equipment, the structural resonance calculator covers plate, shell, and floor-panel geometries beyond the simple beam cases here.

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. Engineers also express damping as the loss factor η = 2ζ (common in structural dynamics) or as logarithmic decrement δ = πη (measured from free-decay ring-down tests). To measure how an actual component responds across a swept frequency range and identify its real resonance peaks experimentally, a vibration frequency analyzer run with a microphone placed near the vibrating part can reveal dominant peaks that match these calculated values.