Calculate the natural frequency of a spring-mass system or a cantilever beam, plus the angular frequency and period. Or convert between frequency, period, and angular frequency directly.
Spring-Mass — f = (1/2π)√(k/m)
Force needed per unit of displacement.
Mass attached to the spring.
Cantilever Beam — first vibration mode
Elastic modulus of the beam material (e.g. 200 GPa for steel).
Second moment of area of the beam's cross-section.
Distributed mass of the beam per meter of length.
Length of the cantilever, fixed at one end.
Convert frequency, period, or angular frequency
Known natural frequency in hertz (cycles per second).
Known time for one full cycle, in seconds.
Known angular frequency in radians per second.
Result
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Enter values above to compute.
4 min read4 steps7 terms3 examples6 FAQsf = (1/2π) √(k/m)
Every mechanical system — a spring holding a mass, a beam bolted to a wall, a bridge deck, a guitar string — has one or more natural frequencies at which it prefers to vibrate when disturbed.
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Walk-through
How to Use This Calculator
4 steps▸
1
Pick a calculation mode
Choose the Spring-Mass tab if you have a discrete mass on a spring (stiffness and mass). Choose Cantilever Beam if you're analyzing a beam fixed at one end (Young's modulus, moment of inertia, mass per length, and length). Choose Period if you already know one of frequency, period, or angular frequency and want the other two.
2
Enter your values
On the Spring-Mass tab, enter stiffness (k) in newtons per meter and mass (m) in kilograms. On the Cantilever Beam tab, enter the material's Young's modulus, the beam's moment of inertia, its mass per unit length, and its length. The calculator updates instantly as you type.
3
Read the result
The result card shows the natural frequency (f) in hertz, along with the angular frequency (ω) in radians per second and the period (T) in seconds. The interpretation line explains the resonance risk in plain language.
4
Convert between frequency, period, and angular frequency
Switch to the Period tab and pick which value you already know — frequency, period, or angular frequency — and the calculator derives the other two using f = 1/T and ω = 2πf.
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Reference
Formula & Methodology
2 formulas▸
Spring-mass system
f = (1/2π) √(k/m)
The natural frequency (f) of an undamped spring-mass system depends on the spring stiffness (k, in newtons per meter) and the attached mass (m, in kilograms). A stiffer spring or a lighter mass both raise the natural frequency.
Cantilever beam (first mode)
f₁ = (λ₁² / 2πL²) √(EI/μ)
The first-mode natural frequency of a uniform cantilever beam (fixed at one end, free at the other) depends on Young's modulus (E), the cross-section's moment of inertia (I), the beam's mass per unit length (μ), and its length (L). λ₁ ≈ 1.875104 is the first-mode eigenvalue of the cantilever boundary-value problem, a standard constant from Euler-Bernoulli beam theory.
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Glossary
Key Terms Explained
7 terms▸
Natural frequency ↗The frequency at which a system oscillates when displaced and released with no external driving force, ignoring damping. Denoted f and measured in hertz (Hz, cycles per second).
Resonance ↗The large-amplitude response that occurs when a system is driven by an external force at (or near) its natural frequency. Avoiding resonance is a key design consideration in mechanical and structural engineering.
Angular frequency ↗The natural frequency expressed in radians per second instead of cycles per second. Related to frequency by ω = 2πf, and used directly in the equations of motion for vibrating systems.
Spring constant ↗Also called stiffness (k) — the force required to displace a spring by one unit of length, measured in newtons per meter (N/m). Larger k means a stiffer spring and a higher natural frequency for a given mass.
Cantilever ↗A beam or structural member fixed (rigidly supported) at one end and free at the other. Cantilever beams are common in structural engineering, from balconies to aircraft wings, and each has its own set of vibration modes.
Vibration mode ↗One of the discrete shapes and frequencies at which a continuous structure (like a beam) naturally oscillates. The first mode is the lowest-frequency, simplest shape; higher modes have progressively higher frequencies and more complex shapes.
Period ↗The time required to complete one full oscillation cycle, denoted T and measured in seconds. Period and frequency are reciprocals: T = 1/f.
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Scenarios
Real-World Examples
3 worked examples▸
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Mechanical engineer
Checking a spring-mounted component
Spring stiffness (k) 1,000 N/mMass (m) 5 kg
f = (1/2π)√(1000/5) = (1/2π)√200 ≈ 2.25 Hz. If the component will experience vibration near 2.25 Hz (for example, from a nearby motor), it's at risk of resonance and should be re-designed or isolated.
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Structural engineer
Estimating a steel cantilever beam's first mode
Young's modulus (E) 200 GPaMoment of inertia (I) 500,000 mm⁴Mass per length (μ) 5 kg/mLength (L) 1 m
f₁ = (1.875²/2π·1²)√(200e9×5e-7/5) ≈ 79.1 Hz. This is the beam's lowest (first-mode) natural frequency — the frequency it would ring at if plucked like a diving board.
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Student
Converting a measured period to frequency
Period (T) 0.2 s
Using the Period tab with "From period": f = 1/T = 1/0.2 = 5 Hz, and ω = 2πf ≈ 31.42 rad/s. This matches a system with a spring stiffness and mass ratio of k/m = (2π×5)² ≈ 987 (s⁻²).
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
Every mechanical system — a spring holding a mass, a beam bolted to a wall, a bridge deck, a guitar string — 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
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For a simple spring-mass system, the natural frequency comes directly from Newton's second law applied to simple harmonic motion: f = (1/2π)√(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, f₁ = (λ₁²/2πL²)√(EI/μ), comes from solving the Euler-Bernoulli beam equation with cantilever (fixed-free) boundary conditions; λ₁ ≈ 1.875104 is the first root of that boundary-value problem, a widely tabulated constant in vibration engineering references.
Inputs and what they mean
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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 — about 200 GPa for structural steel), moment of inertia (I) in quartic millimeters (a cross-section shape property), mass per unit length (μ) in kilograms per meter, and length (L) in meters. On the Period tab, enter whichever single quantity you already know — frequency in hertz, period in seconds, or angular frequency in radians per second — and the calculator fills in the other two using f = 1/T and ω = 2πf.
Limits and edge cases
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Both formulas assume no damping — 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 λ₁. The beam formula also assumes a uniform cross-section along the full length and small deflections within the elastic range — it should be treated as an estimate for tapered, composite, or heavily loaded beams, not a substitute for a full modal analysis.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for natural frequency?+
For a spring-mass system, f = (1/2π)√(k/m), where k is the spring stiffness in N/m and m is the mass in kilograms. For a cantilever beam's first vibration mode, f₁ = (λ₁²/2πL²)√(EI/μ), where λ₁ ≈ 1.875104.
What is resonance and why does it matter?+
Resonance happens when an external force drives a system at (or very close to) its natural frequency, causing the amplitude of vibration to grow much larger than the driving force alone would suggest. Engineers check natural frequency specifically to avoid designing something that will resonate under normal operating conditions (like a motor's running speed or foot traffic on a footbridge).
Does a stiffer spring or beam mean a higher natural frequency?+
Yes. Natural frequency scales with the square root of stiffness (or Young's modulus and moment of inertia, for a beam) and inversely with the square root of mass. A stiffer system vibrates faster; a heavier system vibrates slower, all else equal.
Why does the cantilever beam calculation use a constant like 1.875?+
That value (λ₁ ≈ 1.875104) is the first root of the characteristic equation for a cantilever (fixed-free) beam under the Euler-Bernoulli vibration theory. It's specific to the first (lowest) vibration mode and to fixed-free boundary conditions — other supports (like pinned-pinned) use different tabulated constants.
What units does the Natural Frequency Calculator use?+
Natural frequency is shown in hertz (Hz). Angular frequency is shown in radians per second (rad/s). Period is shown in seconds (s). Spring stiffness is entered in N/m, mass in kg, Young's modulus in GPa, moment of inertia in mm⁴, and mass per length in kg/m.
Does this calculator account for damping?+
No. This version calculates the undamped natural frequency, which is the standard reference value engineers compare a driving frequency against. Real systems have some damping, which slightly lowers the actual resonant frequency and causes free vibrations to decay over time — that effect isn't modeled here.
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