ωn = √(k/m) is the undamped mechanical property. Free oscillation occurs at ωd = ωn√(1−ζ²) when ζ < 1. Under harmonic force, the displacement peak is ωr = ωn√(1−2ζ²) only when ζ < 1/√2. They are not interchangeable.
Underdamped systems oscillate after a disturbance; critically damped systems return without oscillation as quickly as this model allows; overdamped systems return more slowly. Enough damping removes the displacement-resonance peak even though ωn is still defined.
At ideal series resonance, reactances cancel, impedance is minimum and current is maximum. At ideal parallel anti-resonance, branch susceptances cancel, input impedance is maximum and source current is minimum. Loading changes the observed Q.
Convert grams to kilograms, N/mm to N/m, millihenries to henries and microfarads or picofarads to farads before applying formulas. The selectors above do this internally and show the SI substitution in the calculation panel.
Use these results for first-pass vibration isolation, mount selection, filters and tuned circuits. Confirm safety-critical designs with modal testing, frequency-response analysis, component models and tolerances; a single ideal mode cannot represent every physical resonance.
Interpretation: This is underdamped and has a clear forced-response peak close to, but slightly below, 2.013 Hz.
Interpretation: The series current peaks at f0, but Q = 1 gives a broad rather than sharply selective response.
The mechanical calculation is a linear, single-degree-of-freedom model with a massless linear spring and viscous damping. Real spring mass can lower the frequency; nonlinear stiffness makes frequency amplitude-dependent; joints add friction; and structures generally have multiple coupled modes. Operating loads, boundaries and sensor mass also matter.
The electrical calculation uses ideal lumped R, L and C elements. Real inductors and capacitors include ESR, winding resistance, parasitic inductance and capacitance, tolerance, temperature drift and finite self-resonant frequency. Source and load impedance alter Q. At RF, device packages, traces, ground return and PCB layout can dominate the nominal parts. The tolerance output varies L and C independently to their stated extremes; it does not model correlated tolerance, R variation or parasitics.
Undamped natural frequency is a property of mass and stiffness, or inductance and capacitance. Resonant frequency is where a forced steady-state response peaks and can shift or disappear with damping. Damped natural frequency describes free oscillation after a disturbance, so the three terms are not interchangeable.
Yes. For a viscously damped single-degree-of-freedom mechanical system, free oscillation occurs at ωd = ωn√(1−ζ²), while the displacement response peaks at ωr = ωn√(1−2ζ²) only when ζ is below 1/√2. The undamped natural frequency ωn itself remains √(k/m).
Convert mass to kilograms and stiffness to newtons per metre, calculate ωn = √(k/m), then divide by 2π to obtain fn in hertz. This calculator performs those conversions and can rearrange the same equation to solve for mass or stiffness.
You may enter resistance in Ω or kΩ, inductance in H, mH, µH or nH, and capacitance in F, µF, nF or pF. The calculator converts each value to SI internally and scales frequency results from Hz through GHz.
Real springs have mass and nonlinear stiffness, structures have multiple modes, and real components have tolerance, ESR, parasitic reactance, loading and self-resonance. Fixtures, sensors, PCB layout and temperature can also move or broaden the measured peak.
For the ideal series and parallel RLC models used here, Q = f0/Δf and the half-power bandwidth is exact for the modeled topology. High Q means a narrow response. The same simple relation is only an approximation for many coupled, loaded or non-ideal resonators.
An ideal series RLC has minimum impedance and maximum current at f0. An ideal parallel RLC has maximum impedance and minimum source current at its anti-resonance. Series Q falls as series resistance rises; parallel Q rises as shunt resistance rises.
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Verified cases: m = 1 kg and k = 4π² N/m gives fn = 1 Hz; L = 1 H and C = 1/(4π²) F gives f0 = 1 Hz; series R = 10 Ω, L = 10 mH, C = 100 µF gives f0 ≈ 159.155 Hz and Q = 1.
