Learn: Natural circular frequency
A mass on an elastic support can vibrate after being disturbed. Its undamped natural angular frequency describes how fast the vibration phase advances, not the number of full cycles per second.
What the formula is saying
Divide stiffness k by mass m and take the square root: ωn = √(k/m). Stiffness restores displacement while mass resists acceleration; their ratio sets the vibration time scale.
Read the symbols in plain language
- k
- Stiffness
Restoring force per unit displacement for the chosen degree of freedom; it is not an elastic modulus.
N/mUse N/m as the base unit shown here. Use k in N/m and m in kg. k/m has unit s⁻², so its square root is expressed as rad/s. Mass is not weight: a force in N must first be divided by g to obtain kg.
- m
- Mass
Inertial mass of the selected dynamic system, not its weight force. A spectral acceleration times mass produces force.
kgUse kg as the base unit shown here. Use k in N/m and m in kg. k/m has unit s⁻², so its square root is expressed as rad/s. Mass is not weight: a force in N must first be divided by g to obtain kg.
- ωn
- Result to find
Natural circular frequency. A positive mass and stiffness produce a real positive undamped angular frequency.
rad/s
Sort out the units first
Use k in N/m and m in kg. k/m has unit s⁻², so its square root is expressed as rad/s. Mass is not weight: a force in N must first be divided by g to obtain kg.
Assumptions before calculating
Assume a linear, single-degree-of-freedom mass–spring model with positive effective mass and stiffness, small motion and consistent units. The selected mass and stiffness must refer to the same generalized displacement.
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 ωn and explain the result in the stated output unit.
- k · Stiffness
- 1000000 N/m
- m · Mass
- 10000 kg
Compare restoring stiffness with inertia
Stiffness divided by the matching effective mass sets squared natural angular frequency.
(1000000) ÷ (10000) = 100 s⁻²Take the positive square root
A positive mass and stiffness produce a real positive undamped angular frequency.
√((100)) = 10 rad/s
Does this worked answer make sense?
Four times the stiffness doubles ωn; four times the mass halves it. The product of angular frequency and natural period should equal 2π.
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.
- k · Stiffness
- 2000000 N/m
- m · Mass
- 8000 kg
Compare restoring stiffness with inertia
Stiffness divided by the matching effective mass sets squared natural angular frequency.
(2000000) ÷ (8000) = 250 s⁻²Take the positive square root
A positive mass and stiffness produce a real positive undamped angular frequency.
√((250)) ≈ 15.8113883 rad/s
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.
- k · Stiffness
- 1500000 N/m
- m · Mass
- 12000 kg
Find: Learn: Natural circular frequency
A hint, not the answer
Divide stiffness k by mass m and take the square root: ωn = √(k/m). Stiffness restores displacement while mass resists acceleration; their ratio sets the vibration time scale.
Use k in N/m and m in kg. k/m has unit s⁻², so its square root is expressed as rad/s. Mass is not weight: a force in N must first be divided by g to obtain kg.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Compare restoring stiffness with inertia
Stiffness divided by the matching effective mass sets squared natural angular frequency.
(1500000) ÷ (12000) = 125 s⁻²Take the positive square root
A positive mass and stiffness produce a real positive undamped angular frequency.
√((125)) ≈ 11.18033989 rad/s
Avoid the common trap
Do not omit the square root, enter mass in tonnes as if it were kg, or label rad/s as Hz. Divide ωn by 2π to obtain ordinary frequency.
When this method applies — and when it does not
This is the undamped natural frequency of a one-coordinate model, not every mode of a real building or the frequency of a driving force. Multi-degree systems require eigenvalue analysis and nonlinear stiffness changes the relationship.
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 circular 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.