Learn: Natural frequency
Angular frequency counts radians of phase per second, whereas ordinary frequency counts complete cycles per second. One complete cycle contains 2π radians.
What the formula is saying
Divide the supplied angular frequency ω by 2π. This is a unit-and-definition conversion; no new mass or stiffness calculation is required.
Read the symbols in plain language
- ωn
- Natural circular frequency
Undamped angular frequency in radians per second. Divide by 2π to obtain cycles per second, not by π alone.
rad/sUse rad/s as the base unit shown here. ω must be in rad/s and f is in Hz, meaning cycles per second. Degrees per second and revolutions per minute require their own conversions before this formula can be used.
- fn
- Result to find
Natural frequency. Dividing radians per second by radians per cycle leaves cycles per second.
Hz
Sort out the units first
ω must be in rad/s and f is in Hz, meaning cycles per second. Degrees per second and revolutions per minute require their own conversions before this formula can be used.
Assumptions before calculating
Assume the supplied angular frequency describes the same oscillation whose cyclic frequency is wanted. A natural, damped or forcing angular frequency can be converted, but its physical label must be retained.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find fn and explain the result in the stated output unit.
- ωn · Natural circular frequency
- 10 rad/s
Write the radians in one complete cycle
A full oscillation advances phase by two pi radians.
2 × π ≈ 6.283185307 rad/cycleConvert phase rate to cycles per second
Dividing radians per second by radians per cycle leaves cycles per second.
(10) ÷ (6.283185307) ≈ 1.591549431 Hz
Does this worked answer make sense?
An angular frequency of 2π rad/s corresponds to 1 Hz. Converting back with ω = 2πf should reproduce the original value.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- ωn · Natural circular frequency
- 15 rad/s
Write the radians in one complete cycle
A full oscillation advances phase by two pi radians.
2 × π ≈ 6.283185307 rad/cycleConvert phase rate to cycles per second
Dividing radians per second by radians per cycle leaves cycles per second.
(15) ÷ (6.283185307) ≈ 2.387324146 Hz
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- ωn · Natural circular frequency
- 8 rad/s
Find: Learn: Natural frequency
A hint, not the answer
Divide the supplied angular frequency ω by 2π. This is a unit-and-definition conversion; no new mass or stiffness calculation is required.
ω must be in rad/s and f is in Hz, meaning cycles per second. Degrees per second and revolutions per minute require their own conversions before this formula can be used.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Write the radians in one complete cycle
A full oscillation advances phase by two pi radians.
2 × π ≈ 6.283185307 rad/cycleConvert phase rate to cycles per second
Dividing radians per second by radians per cycle leaves cycles per second.
(8) ÷ (6.283185307) ≈ 1.273239545 Hz
Avoid the common trap
Do not multiply by 2π when converting rad/s to Hz; that is the reverse conversion. Do not treat a numerical angular frequency of 10 rad/s as 10 cycles/s.
When this method applies — and when it does not
The conversion does not establish whether resonance occurs or which structural mode is active. Zero angular frequency converts to zero Hz; a period calculated as 1/f would then be undefined.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Natural frequency. Single-degree-of-freedom free vibration, natural frequency, period and viscous damping. Angular frequency and cycles per second are different quantities.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.