Dynamics · Vibration
Resonance Calculator
Size natural frequency and forced response for a spring-mass-damper — watch the frequency-response curve track your inputs in real time.
Live frequency-response preview
Result
Natural frequency
fn = (1/2π)√(k/m)
Enter mass and spring stiffness — the diagram plots the frequency-response curve live.
Spring-mass-damper fundamentals — natural frequency, forced response, and damping.
A single-degree-of-freedom (SDOF) system has mass m, spring stiffness k, and viscous damping c. The equation of motion under harmonic forcing is:
m·ẍ + c·ẋ + k·x = F(t)
- SpringRestoring force F = −kx — stiffness k in N/m
- MassInertia F = ma — mass m in kg
- DamperVelocity drag F = −cẋ — coefficient c in N·s/m
- Natural frequencyfn = (1/2π)√(k/m)
- Damping ratioζ = c / (2√(km))
- Critical dampingcc = 2√(km)
- Damped natural freq.fd = fn·√(1−ζ²)
- Amplitude factorM = 1/√[(1−r²)² + (2ζr)²]
- Phase angleφ = tan⁻¹(2ζr / (1−r²))
- Resonant frequencyfr = fn·√(1−2ζ²) (ζ < 0.707)
- Natural periodτ = 1/fn
| Case | Behavior |
|---|---|
| Undamped (ζ = 0) | Oscillates indefinitely — idealized, no energy loss. |
| Underdamped (0 < ζ < 1) | Decaying oscillations — most real machinery. |
| Critically damped (ζ = 1) | Fastest non-oscillatory return — door closers, gauges. |
| Overdamped (ζ > 1) | Slow decay without oscillation. |
Underdamped (ζ < 1)
x(t) = A·e^(−ζωn·t)·sin(ωd·t + φ)
Decaying oscillations; ωd = ωn·√(1−ζ²)
Critically damped (ζ = 1)
x(t) = (A + Bt)·e^(−ωn·t)
Fastest return without overshoot
Overdamped (ζ > 1)
x(t) = A₁·e^(r₁t) + A₂·e^(r₂t)
Non-oscillatory, slower than critical
Forced at resonance
M → max when r ≈ 1
Response lags forcing by 90° at peak
Engineering applications
- Vibration isolation in machinery
- Automotive suspension tuning
- Seismic base isolation
- Tuned mass dampers in tall buildings
- Rotating equipment critical speed analysis
Design considerations
- Keep operating frequency away from fn
- Add damping to limit resonant peaks
- Isolation mounts: mount fn ≪ forcing frequency
- Use critical damping for fast positioning without overshoot
For transient time-domain analysis, see the Damping Calculator.
- Tacoma Narrows Bridge (1940) — classic resonance failure from wind excitation.
- Tuned mass dampers atop skyscrapers shift fn away from wind or seismic forcing.
- Automotive engine mounts are tuned to keep idle vibration out of the cabin.
- At resonance with light damping, response lags the forcing by 90°.
- Critical damping cc = 2√(km) is the threshold between oscillatory and aperiodic motion.
- Isolation mounts work best when the mount fn is well below the disturbing frequency.
Worked example: Natural frequency of a 1 kg mass on a 1000 N/m spring
- fn = (1/2π)√(k/m) = (1/2π)√(1000/1)
- ωn = √(1000) ≈ 31.6 rad/s
fn ≈ 5.03 Hz
What is the natural frequency?
The natural frequency is the rate at which a spring-mass system oscillates with no external forcing, fn = (1/2π)√(k/m). Driving the system near this frequency causes resonance and large vibration amplitudes.
What is the amplitude magnification factor?
It is the ratio of the dynamic (vibrating) amplitude to the static deflection: M = 1/√[(1−r²)² + (2ζr)²], where r = forcing/natural frequency. At resonance with light damping M can become very large.
Why is there sometimes no resonant frequency?
A distinct resonant peak only exists for damping ratios below 1/√2 (≈ 0.707). For more heavily damped systems the response simply decreases as frequency rises, so the calculator reports no resonant peak.
What does the damped natural frequency mean?
For underdamped systems the free oscillation happens at fd = fn√(1−ζ²), slightly below the natural frequency. Critically and over-damped systems do not oscillate, so a damped natural frequency is not defined.