Natural Frequency Calculator
A spring–mass system vibrates at ωn = √(k/m): 10 kg on a 4000 N/m spring has ωn = 20 rad/s, fn = 3.1831 Hz.
0 for an undamped system.
3.18309886183791 Hz
Full answer, then cut to fewer decimals (not rounded):
- 3.18309886183791 Hz
- 3.1830988618379 Hz
- 3.183098861837 Hz
- 3.18309886183 Hz
- 3.1830988618 Hz
- 3.183098861 Hz
- 3.18309886 Hz
- 3.1830988 Hz
- 3.183098 Hz
- 3.18309 Hz
- 3.1830 Hz
- 3.183 Hz
- 3.18 Hz
- 3.1 Hz
- Angular natural frequency ωn = √(k/m)
- 20 rad/s
- Period T = 1/fn
- 0.314159 s
- Static deflection δ = mg/k
- 24.525 mm
- Critical damping cc = 2√(km)
- 400 N·s/m
- Damping ratio ζ = c/cc
- 0
- Damping
- undamped
Step-by-step solution
- ωn = √(k/m) = √(4000 ÷ 10) = 20 rad/s
- fn = ωn/2π = 3.1831 Hz
- ζ = c / 2√(km) = 0 ÷ 400 = 0
Natural Frequency Formula
ωn = √(k/m), fn = ωn/2π, ζ = c/2√(km)
m mass, k stiffness, c damping coefficient
Worked Example
A 10 kg mass on a 4000 N/m spring:
- ωn = √(k/m) = √(4000 ÷ 10) = 20 rad/s
- fn = ωn/2π = 3.1831 Hz
- ζ = c / 2√(km) = 0 ÷ 400 = 0
Period T = 1/fn: 0.314159 s · Static deflection δ = mg/k: 24.525 mm · Critical damping cc = 2√(km): 400 N·s/m · Damping ratio ζ = c/cc: 0 · Damping: undamped
More Theory of Machines and Vibrations Calculators
-
Transmissibility Calculator
TR = √[1 + (2ζr)²] ÷ √[(1 − r²)² + (2ζr)²]
r = ω/ωn frequency ratio, ζ damping ratio
Example: A machine running at twice the natural frequency with ζ = 0.1: Transmissibility TR = 0.355862.
-
Critical Speed Calculator
ωc = √(g/δ) = √(k/m), Nc = 60ωc/2π
δ static deflection under the rotor, g = 9.81 m/s²
Example: A rotor whose weight deflects the shaft 0.5 mm: Critical (whirling) speed Nc = 1337.58 rpm.
-
Gear Train Calculator
N₁/Nout = Π driven teeth ÷ Π driving teeth
train value = 1/speed ratio
Example: A compound train 20/60 then 15/45 driven at 1200 rpm: Output speed N = 133.333 rpm.
-
Flywheel Calculator
ΔE = I ω² Cs = m k² ω² Cs
Cs = (ωmax − ωmin)/ω, k radius of gyration
Example: ΔE = 5000 J at 600 rpm with Cs = 2 % and k = 0.5 m: Moment of inertia I = 63.3257 kg·m².
Frequently asked questions
How do I find natural frequency from static deflection?
ωn = √(g/δ), because k/m = g/δ. A static deflection of 1 mm gives about 15.8 Hz.
What is the logarithmic decrement?
The natural log of the ratio of successive amplitudes, δ = 2πζ/√(1 − ζ²); it is used to measure damping from a free-vibration record.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Natural Frequency Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/natural-frequency-calculator/
IEEE
Scientific Converters, "Natural Frequency Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/natural-frequency-calculator/