Spring-Mass Resonance Calculator
This spring-mass resonance calculator finds the natural frequency from spring stiffness k and mass m, adds a damping ratio to get the damped frequency, plots the force transmissibility versus frequency ratio, sizes a mount for vibration isolation, and solves a two-degree-of-freedom (2-DOF) coupled system from its eigenvalues.
ℹ These are the exact textbook formulas for an idealised linear lumped-parameter system: point mass, a massless linear spring of constant stiffness, viscous (velocity-proportional) damping, and rigid mounting. Real mounts are nonlinear, have mass and internal resonances, and damping is rarely purely viscous — so treat these as a design starting point, then verify with a measurement. Results are only as good as the k, m and damping values you enter; check them against your spring rate and component datasheets.
Inputs
Force transmissibility vs frequency ratio r = f / fn
TR > 1 = amplified, TR < 1 = isolated. Isolation begins past r = √2 ≈ 1.414.
Oscillation animation
A free oscillation of the mass on its spring at the damped natural frequency. With ζ > 0 the swing decays; at ζ ≥ 1 it does not oscillate. This is a schematic, not to scale.
Inputs
Two masses in a chain: ground —k₁— m₁ —k₂— m₂ (free end). The two natural frequencies are the square roots of the eigenvalues of M⁻¹K. Undamped.
Inputs
How It Works
A mass m on a linear spring of stiffness k has one undamped natural frequency: fn = (1/2π)·√(k/m), in hertz, where the angular natural frequency is ωn = √(k/m) in rad/s. This is the rate at which the mass freely bounces if you displace it and let go — the fundamental result of a single-degree-of-freedom (SDOF) lumped-parameter model. Add a damping ratio ζ (the actual damping divided by the critical damping cc = 2√(km)) and the frequency you actually observe in free vibration drops slightly to the damped natural frequency fd = fn·√(1−ζ²). When ζ < 1 the system is underdamped and oscillates; at ζ = 1 it is critically damped and returns without overshoot; above 1 it is overdamped. A quick shortcut: fn = (1/2π)·√(g/δst), where δst = mg/k is the static deflection under gravity — useful when it is easier to measure sag than to know k directly.
For vibration isolation, the key quantity is force transmissibility — the fraction of a harmonic force (or base motion) that passes through the mount. As a function of the frequency ratio r = f/fn, TR = √(1 + (2ζr)²) / √((1−r²)² + (2ζr)²). At r = 1 the system is at resonance and TR peaks (the lower the damping, the taller the peak). Crucially, TR = 1 at r = √2 ≈ 1.414 regardless of damping, and isolation only happens for r > √2 — the disturbance frequency must sit comfortably above the system’s natural frequency. A typical design target is r ≥ 3, giving 90% isolation or better even with modest damping. To isolate a known disturbance to a target TR, you make fn low enough (softer spring or more mass) that r is large; the isolation tab inverts the undamped relation to find the maximum isolator stiffness that achieves your target — so the required mount stiffness (k) can be read directly from the result. Anti-vibration mounts used under HVAC compressors, generators, and precision optical tables are sized exactly this way. For related acoustic isolation problems, the room mode calculator addresses standing-wave resonances that occur when a room’s dimensions match sound wavelengths.
For a two-degree-of-freedom chain — ground–k₁–m₁–k₂–m₂ — there are two natural frequencies, often called eigenfrequencies or normal mode frequencies. The undamped equations of motion give a stiffness matrix K and a mass matrix M; the squared angular frequencies are the eigenvalues of M⁻¹K, which for a 2×2 system come from solving a quadratic exactly (no iteration). The lower mode has both masses moving in phase; the higher mode has them moving out of phase. This is the basis of the tuned-mass damper (TMD) idea, where a small secondary mass on a spring is deliberately tuned to split and absorb energy from a troublesome primary resonance — widely used in tall buildings, bridges, and precision machine tool spindles. For more complex multi-DOF systems and continuous structures, the structural resonance calculator covers beam and string mode shapes.