Natural angular frequency
Undamped natural angular frequency in radians per second.
Physics · Mechanics · Motion & Energy
Spring–mass natural frequency calculator
Find undamped natural angular frequency, frequency and period from moving mass and spring stiffness. This describes oscillation about equilibrium.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Natural frequency
Number of complete cycles per second.
Period
Duration of one complete cycle.
Relation and numerical substitution
ωₙ = √(k / m)—
Ideal undamped linear spring; m,k > 0; oscillations about equilibrium.
01Formulas and symbols
Formulas used
ωₙ
ωₙ = √(k / m)Undamped natural angular frequency in radians per second.fₙ
fₙ = ωₙ / (2π)Number of complete cycles per second.T
T = 1 / fₙDuration of one complete cycle.02Assumptions and limits
Scope of validity
- Ideal undamped linear spring; m,k > 0; oscillations about equilibrium.
03Validation example
Reference numerical case
- m = 2 kg, k = 200 N/m: ωₙ = √(200/2) = 10 rad/s; fₙ = 10/(2π) ≈ 1.59154943 Hz; T ≈ 0.62831853 s.
04References
Technical references
FAQ
Does gravity change the natural frequency of a vertical spring–mass system?
In the ideal linear one-degree-of-freedom model, gravity shifts the static equilibrium position but does not change ωₙ = √(k/m) for small oscillations about that equilibrium.
What happens to natural frequency if stiffness or mass is doubled?
Because fₙ is proportional to √(k/m), doubling stiffness multiplies the frequency by √2. Doubling the mass divides the frequency by √2.
05How it works
The spring restoring force accelerates the moving mass. Increasing stiffness raises natural frequency; increasing mass lowers it. Quadrupling stiffness doubles frequency, while quadrupling mass halves it. Frequency in Hz counts cycles; angular frequency in rad/s equals 2π times that value.
06Limits of the model
A single translational degree of freedom, linear massless spring, positive moving mass and no damping. Distributed spring mass, damping, multiple coupled masses and nonlinear stiffness need another model. No resonance amplitude or forced-response prediction is made.
07Common mistakes
1 N/mm is 1000 N/m. Use the moving mass, not its weight in newtons. Do not confuse natural angular frequency with cycles per second.