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Critical Damping & Natural Frequency

Find critical viscous damping, mass, stiffness and the undamped natural frequency of a linear spring–mass system.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A 1 kg mass on a 1 N/m spring requires 2 N·s/m of critical damping. Its undamped natural frequency is 1/(2π) Hz, approximately 0.159155 Hz.

Critical damping c = 2√(k m); undamped natural frequency f = √(k/m)/(2π) = c/(4πm)

Worked example

Enter these known values and leave the other values blank.

Oscillating mass
1 kg
Linear spring stiffness
1 N/m
Undamped natural frequency f
0.15915 Hz
Critical viscous damping coefficient
2 N·s/m

Assumptions and limitations

  • Mass, stiffness, critical damping and frequency must be strictly positive. The model is one linear degree of freedom with viscous force proportional to velocity.
  • Frequency in hertz counts cycles per second. Angular frequency ω equals 2πf in radians per second. RPM is a cycles-per-minute expression of the same frequency.
  • The displayed frequency is the undamped system’s natural frequency. A critically damped free response does not oscillate at that frequency. Initial displacement and velocity, response time and forcing are not modeled.

Common questions

How does critical damping change when mass quadruples?

At fixed spring stiffness, critical damping doubles and undamped natural frequency halves.

Is this a damping ratio?

No. This result is a dimensional damping coefficient. Damping ratio is actual damping divided by the critical coefficient; it equals one at critical damping.

References

Bookify distinguishes ordinary frequency in hertz from angular frequency, correcting the captured missing factor of 2π. Pound-force stiffness and damping units use the exact international pound and length with standard gravity.

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