Natural Frequency Calculator

Spring–mass, pendulum, and cantilever beam natural frequencies. Solves f = (1/2π)√(k/m) in any direction, plus ω and period. Free, client-side, no signup.

Formulas & method

Spring–mass:

ω = √(k/m) · f = ω/2π · T = 1/f

Pendulum (small angles):

f = (1/2π)√(g/L)

Cantilever beam, first bending mode (Euler–Bernoulli):

f₁ = (β₁L)²/(2πL²) · √(EI/ρA), β₁L = 1.8751

Assumes a uniform rectangular section (I = wh³/12), a perfectly clamped root, small deflections, and no damping, shear deformation, or tip mass. Results are estimates for design intuition, not a substitute for full modal analysis.

Frequently asked questions

What is the natural frequency of a system?
The natural frequency is the rate at which a system vibrates on its own once disturbed and left alone, with no external force driving it. For a spring-mass system it depends only on the spring stiffness k and the mass m: f₀ = (1/2π)√(k/m). Damping is ignored in this ideal definition; real systems vibrate slightly slower.
What is the natural frequency formula for a spring and mass?
f = (1/2π)√(k/m), where k is the spring stiffness in N/m and m is the mass in kg. The angular frequency is ω = √(k/m) rad/s and the period is T = 2π√(m/k) seconds. Example: k = 100 N/m and m = 1 kg gives f = (1/2π)√(100/1) ≈ 1.5915 Hz. Doubling the stiffness raises the frequency by √2; doubling the mass lowers it by √2.
How do I find the spring stiffness or mass from a target frequency?
Rearrange the formula. For a target frequency f and known mass m: k = (2πf)²·m. For a target frequency and known stiffness k: m = k/(2πf)². This tool's spring-mass mode solves any direction: give any two of frequency, stiffness, and mass, and it computes the third plus ω and the period.
What is the natural frequency of a pendulum?
f = (1/2π)√(g/L), where g is gravity (9.80665 m/s² on Earth) and L is the pendulum length in meters. The mass of the bob does not appear: a heavier and a lighter bob swing at the same rate for small angles. Example: a 1 m pendulum on Earth gives f ≈ 0.4984 Hz, i.e. a period of about 2.006 s.
What is the natural frequency of a cantilever beam?
For the first bending mode of a uniform rectangular cantilever, f₁ = (β₁L)²/(2πL²)·√(EI/ρA) with (β₁L) = 1.8751, I = wh³/12 the second moment of area, and A = wh the cross-section. Example: a 1 m steel strip 5 cm wide × 1 cm thick (E = 200 GPa, ρ = 7850 kg/m³) has f₁ ≈ 8.15 Hz. Higher modes (2.7×β² etc.) are not computed here.
What assumptions does this calculator make?
Four main ones. (1) No damping: real friction and air resistance lower the observed frequency slightly. (2) Linear springs and small deflections: Hooke's law must hold. (3) Small pendulum angles: the formula drifts above roughly 15° of swing. (4) For the beam: a perfectly clamped root, uniform rectangular section, L much larger than thickness (Euler-Bernoulli neglects shear and rotary inertia), and no tip mass.
What is the difference between f, ω, and T?
f is the cyclic frequency in hertz (cycles per second). ω = 2πf is the angular frequency in radians per second, the form that appears directly in the equations of motion. T = 1/f is the period in seconds, the time for one full cycle. This tool reports all three so you can use whichever your equation needs.
Why does natural frequency matter? What is resonance?
When an external force pushes a system at or near its natural frequency, each push adds energy in phase and the vibration amplitude grows rapidly - that is resonance. Bridges, buildings, aircraft parts, and machine mounts are all designed so their natural frequencies stay clear of expected excitation frequencies. A quick estimate with this calculator is the first step before a full dynamic analysis.
Is this the same as a frequency unit converter?
No. This tool computes the natural frequency of a physical system (spring-mass, pendulum, or beam) from its properties. A frequency converter only changes units, e.g. Hz to kHz or MHz. If you already have a frequency and just want it in different units, use our frequency-converter tool instead.

Free tool by NRTechStudio

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