Spring Natural Frequency Calculator
Enter the spring constant and mass to find the natural frequency.
Result
How it is calculated
The tool treats the spring and mass as an ideal undamped oscillator. The angular frequency is ω = √(k ÷ m) in rad/s, the natural frequency is f = ω ÷ (2π) in hertz, and the period is T = 1 ÷ f in seconds. Here k is the spring constant in N/m and m the moving mass in kg; the formulas assume these SI units, so convert grams or lb/in before entering them. Both values must be greater than zero, otherwise no result is shown. Frequency is displayed to 3 decimals, ω to 2 and the period to 4. The tool ignores damping and the spring's own mass, so a heavy spring or strong friction will make a real system oscillate a little slower or die out.
Worked example
A spring with k = 200 N/m carries a 0.5 kg mass. First ω = √(200 ÷ 0.5) = √400 = 20.00 rad/s. Then f = 20 ÷ (2π) = 20 ÷ 6.2832 = 3.183 Hz. The period is T = 1 ÷ 3.183 = 0.3142 s, so the mass completes about three full bounces per second. Quadrupling the mass to 2 kg would halve the frequency to about 1.59 Hz, because frequency depends on the square root of k ÷ m.
Common mistakes
- Entering the mass in grams. A 500 g mass must be typed as 0.5; entering 500 gives 0.101 Hz instead of 3.183 Hz, a result about 31 times too low.
- Using a spring rate in N/mm or lb/in without converting. The formula expects N/m, so 2 N/mm must be entered as 2,000.
- Mixing up hertz and rad/s. A figure of 20 rad/s is only about 3.18 cycles per second; quoting ω as the frequency overstates it by 2π.
- Expecting the result to include the weight of the spring itself, or air and friction losses. The tool only uses the attached mass, so a spring that is heavy compared with the load gives a frequency slightly too high.
- Applying it to a pendulum or a beam. The pendulum depends on length and gravity, not k and m, and a beam has its own stiffness formula.
About this calculator
Every undamped mass-spring system oscillates at a characteristic natural frequency, given by f = (1/2π)√(k/m), where k is the spring constant and m is the attached mass. This calculator derives that frequency along with the angular frequency ω = √(k/m) and the oscillation period T = 1/f, the three interlinked quantities that describe how fast the system bounces once displaced and released.
Natural frequency is a foundational concept in mechanical and structural engineering, used in vibration analysis to predict resonance — the dangerous amplification that occurs when an external driving force matches a system's natural frequency — and in the design of suspensions, vibration isolators, and precision instruments. Mechanical engineers use it early in design to keep a component's natural frequency safely away from expected operating or excitation frequencies.
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