What is the formula for natural frequency?

For an undamped single-degree-of-freedom spring-mass system, the angular natural frequency is ω_n = sqrt(k / m) in rad/s, and the cyclic natural frequency is f_n = ω_n / (2π) in Hz. Here k is the spring stiffness in N/m and m is the mass in kg.

What is the difference between Hz and rad/s?

Hz (Hertz) counts complete oscillation cycles per second and is the unit most used in specifications and measurements. Radians per second (rad/s) is the angular form; one full cycle equals 2π radians, so f (Hz) = ω (rad/s) / (2π).

Does damping affect the natural frequency?

Yes, but only slightly for typical engineering structures. The damped natural frequency is ω_d = ω_n × sqrt(1 − ζ²), where ζ is the damping ratio. For common structural damping (ζ < 0.1, i.e., under 10%), the difference from the undamped value is less than 0.5% and is often ignored in preliminary design.

What happens if I enter zero or negative values?

The calculator will display an inline error message. Stiffness and mass must both be positive numbers. Zero stiffness means no restoring force (the system cannot oscillate), and zero or negative mass is physically meaningless.

How do I convert lbf/in to N/m?

1 lbf/in = 175.127 N/m. To convert, multiply your lbf/in value by 175.127. This calculator handles unit conversion automatically — simply select the correct unit from the dropdown.

Can this calculator handle multiple degrees of freedom (MDOF) systems?

No. This tool is for single-degree-of-freedom (SDOF) systems only. MDOF systems require eigenvalue analysis (matrix methods) to find multiple mode shapes and frequencies. Use finite-element software for those cases.

Modal Frequency Calculator

Enter spring stiffness and mass to instantly compute the undamped natural frequency of a single-degree-of-freedom system in Hz and rad/s.

System Parameters

Spring Stiffness (k)

Stiffness of the spring or structural element.

Mass (m)

Total mass of the vibrating body.

Natural Frequency

Enter stiffness and mass, then click Calculate

Typical Natural Frequency Ranges

System	Typical f_n (Hz)	Notes
Building floor (civil)	2 – 8	Walking excitation concern below 8 Hz
Automotive suspension	1 – 3	Ride comfort target range
Machine tool spindle	100 – 1000	Must exceed operating speed range
Vibration isolator mount	3 – 15	Keep < 1/√2 of forcing frequency

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Summary

This modal frequency calculator computes the undamped natural (modal) frequency of a single-degree-of-freedom (SDOF) spring-mass system using the formula f_n = (1 / 2π) × sqrt(k / m), where k is the spring stiffness and m is the system mass. Results are shown simultaneously in Hertz (Hz) and radians per second (rad/s) — the two standard representations used in structural dynamics, mechanical vibration analysis, and engineering design. All calculations run entirely in your browser with no data sent to any server.

How it works

Enter the spring stiffness (k) with its unit (N/m, kN/m, lbf/in, or lbf/ft).
Enter the mass (m) with its unit (kg, g, lbm, or slug).
Click Calculate to compute the natural frequency instantly.
Results are displayed as angular frequency (rad/s) and cyclic frequency (Hz).
Use the Reset button to clear inputs and start a new calculation.

Use cases

Predicting resonance conditions in mechanical assemblies before prototyping.
Sizing suspension springs and masses for target natural frequencies.
Verifying structural dynamic models against hand-calculation benchmarks.
Selecting isolator stiffness to avoid resonance with excitation frequencies.
Teaching vibration fundamentals in undergraduate engineering courses.
Checking equipment mounting designs against machine operating frequencies.
Quick sanity-checking finite-element modal analysis results.
Estimating natural frequency shifts when mass or stiffness is modified.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu