Spring-Mass Natural Frequency Calculator

Every mass-spring system has a natural frequency: the rate at which it oscillates when disturbed from rest with no external forcing. Understanding this frequency is essential for avoiding resonance, designing vibration isolators, and modeling mechanical and structural systems.

Natural angular frequency (undamped):

omega_n = sqrt(k / m) [rad/s]

Natural frequency:

f_n = omega_n / (2 x pi) = (1 / 2pi) x sqrt(k / m) [Hz]

Period of oscillation:

T = 1 / f_n [seconds]

With damping. Real systems have energy dissipation from friction, material damping, or fluid resistance. The damping ratio zeta (greek letter) characterizes how quickly oscillations decay:

zeta = c / (2 x sqrt(k x m))

where c is the viscous damping coefficient in N s/m.

• zeta < 1: underdamped (system oscillates with decaying amplitude)
• zeta = 1: critically damped (returns to equilibrium as fast as possible without oscillating)
• zeta > 1: overdamped (returns slowly without oscillating)

Damped natural frequency (valid for zeta < 1):

omega_d = omega_n x sqrt(1 - zeta^2)

Why resonance is dangerous. If an external force drives the system at exactly f_n with no damping, amplitude grows without bound. The Tacoma Narrows Bridge collapse in 1940 is the classic engineering example. Real systems always have some damping, but resonance still causes dramatically amplified vibration. Always check that operating frequencies stay well away from f_n.

A useful intuition about the square root. Because f_n depends on sqrt(k/m), the relationship is sublinear in both directions. Doubling the spring stiffness increases the frequency by only sqrt(2), about 41%, not 100%. Halving the mass also gives only a 41% increase. To double the natural frequency, you need to quadruple the stiffness or cut the mass to one-quarter. This is why a small change in damping or trim weight rarely shifts a system’s frequency by much in practice.

Beyond the spring-mass model. Real distributed structures (beams, plates, shafts) have an infinite series of natural frequencies, not just one. For a simply-supported beam of length L:

f_n = (n² · π / (2L²)) · sqrt(EI / (ρ · A))

where n is the mode number (1, 2, 3, …), E is Young’s modulus, I is the area moment of inertia of the cross-section, ρ is density, and A is the cross-sectional area. The first mode (n = 1) is the fundamental frequency. Higher modes have higher frequencies and more complex deflection shapes (the beam bends with multiple “humps”). For most engineering work, the fundamental is what matters: it’s the easiest to excite and almost always governs design.