Spring-Mass Natural Frequency Calculator
Find the natural frequency, angular frequency, and resonant period of a spring-mass system.
Add damping to compute damped natural frequency and damping ratio.
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.