Get critical damping values for design and tuning. Choose inputs and units easily. See results instantly above, then download reports fast.
| Case | m (kg) | k (N/m) | ωₙ (rad/s) | fₙ (Hz) | c₍c₎ (N·s/m) |
|---|---|---|---|---|---|
| Light suspension | 2.0 | 800 | 20.000 | 3.183 | 80.000 |
| Machine mount | 12.5 | 1500 | 10.954 | 1.743 | 273.861 |
| Precision stage | 0.8 | 5000 | 79.057 | 12.581 | 126.491 |
Values are rounded for readability. Your results may differ slightly with full precision.
Critical damping (ζ = 1) is the boundary between oscillatory and non-oscillatory behavior. It returns to equilibrium fast without overshoot in ideal second-order models.
Tip: If your damping ratio is near one, small measurement errors matter.
Many vibration and motion-control problems behave like a second-order system with inertia, stiffness, and damping. The critical damping coefficient is the reference value where oscillation just disappears. This calculator computes that reference for translational and rotational models, and can also report the damping ratio. It also helps you validate units and compare design targets before you export reports.
The damping ratio is ζ = c/c₍c₎. When ζ < 1, the response oscillates. When ζ = 1, it is critically damped. When ζ > 1, it is overdamped and returns without oscillation.
For translation, the model is m ẍ + c ẋ + k x = 0. Here c is your actual damper value. The calculator finds c₍c₎ from m and k, then compares them if you provide c.
The relationships are ωₙ = √(k/m) and c₍c₎ = 2 √(k m). With m = 12.5 kg and k = 1500 N/m, ωₙ ≈ 10.954 rad/s and c₍c₎ ≈ 273.861 N·s/m. If c = 180 N·s/m, then ζ ≈ 0.657 (underdamped).
If you know the natural frequency, use the frequency mode. Since c₍c₎ = 2 m ωₙ, m = 2.0 kg and ωₙ = 20 rad/s gives c₍c₎ = 80 N·s/m. This avoids estimating stiffness from geometry.
For torsional motion, use J θ̈ + c θ̇ + Kₜ θ = 0. Replace mass with inertia J and stiffness with torsional stiffness Kₜ. Then ωₙ = √(Kₜ/J) and c₍c₎ = 2 √(Kₜ J), in N·m·s/rad.
Many mounts and isolators operate around ζ ≈ 0.2–0.7 to balance overshoot and settling. Precision positioning and control often use higher ratios, sometimes approaching ζ ≈ 0.7–1.0.
The critical coefficient scales with the square root: c₍c₎ ∝ √m for fixed stiffness and c₍c₎ ∝ √k for fixed mass. Doubling mass or stiffness increases c₍c₎ by about √2 ≈ 1.414.
Translational damping is N·s/m, while rotational damping is N·m·s/rad. A quick check is c₍c₎ ≈ 2 m ωₙ after computing ωₙ from k/m. If results look 1000× off, stiffness units like N/mm versus N/m are a common cause.
It is the damping value that makes a second-order system return to equilibrium as fast as possible without oscillating, under ideal linear assumptions.
c is your actual damper value. c₍c₎ is the computed reference for ζ = 1. Their ratio ζ = c/c₍c₎ indicates under, critical, or over damping.
Use N·s/m in SI. If your damper is specified in lbf·s/ft or N·s/mm, select the matching unit so the calculator converts it correctly.
Yes. If you know ωₙ or fₙ, the calculator uses c₍c₎ = 2 m ωₙ and derives k = m ωₙ² for the translational case.
Because the classification depends on ζ, not raw c. If mass or stiffness is large, c₍c₎ can be very large too. Compare with ζ to judge damping strength properly.
Mass becomes inertia J, stiffness becomes torsional stiffness Kₜ, and damping uses N·m·s/rad. The same formulas apply with rotational variables.
It is a useful reference, but real systems can be nonlinear, multi-mode, or frequency-dependent. Use c₍c₎ as a baseline, then validate with measurements or simulations for your operating range.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.