Every mechanical system โ a spring holding a mass, a beam bolted to a wall, a bridge deck, a guitar string โ has one or more natural frequencies at which it prefers to vibrate when disturbed. Knowing that frequency matters because driving a system near it causes resonance, which can amplify vibration to damaging levels. This calculator finds the natural frequency for two common cases (a spring-mass system and a cantilever beam) and converts freely between frequency, angular frequency, and period.
How the Natural Frequency Calculator works
For a simple spring-mass system, the natural frequency comes directly from Newton's second law applied to simple harmonic motion: f = (1/2ฯ)โ(k/m). A stiffer spring (higher k) raises the frequency; a heavier mass (higher m) lowers it. For a continuous structure like a cantilever beam, the same idea applies but the beam's own stiffness and distributed mass replace the discrete spring and point mass. The first-mode formula, fโ = (ฮปโยฒ/2ฯLยฒ)โ(EI/ฮผ), comes from solving the Euler-Bernoulli beam equation with cantilever (fixed-free) boundary conditions; ฮปโ โ 1.875104 is the first root of that boundary-value problem, a widely tabulated constant in vibration engineering references.
Inputs and what they mean
Spring stiffness (k) is entered in newtons per meter and mass (m) in kilograms. For the cantilever beam, Young's modulus (E) is entered in gigapascals (a material property โ about 200 GPa for structural steel), moment of inertia (I) in quartic millimeters (a cross-section shape property), mass per unit length (ฮผ) in kilograms per meter, and length (L) in meters. On the Period tab, enter whichever single quantity you already know โ frequency in hertz, period in seconds, or angular frequency in radians per second โ and the calculator fills in the other two using f = 1/T and ฯ = 2ฯf.
Limits and edge cases
Both formulas assume no damping โ real systems lose some energy to friction and material damping, which slightly lowers the observed frequency and causes oscillations to decay over time; this calculator does not model that decay. The cantilever formula only returns the first (lowest) vibration mode; a real beam has infinitely many higher modes at higher frequencies, each requiring a different eigenvalue in place of ฮปโ. The beam formula also assumes a uniform cross-section along the full length and small deflections within the elastic range โ it should be treated as an estimate for tapered, composite, or heavily loaded beams, not a substitute for a full modal analysis.