Spring-Mass Resonance Calculator
Find the natural frequency of a spring-mass system from the spring stiffness k and mass m, add a damping ratio to get the damped frequency, plot the force transmissibility versus frequency ratio, size a mount for vibration isolation, and solve 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. 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.
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. 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 tool inverts the undamped relation to find the maximum stiffness that achieves your target.
For a two-degree-of-freedom chain — ground–k₁–m₁–k₂–m₂ — there are two natural 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 idea, where a small second mass is tuned to split and tame a troublesome resonance.