Critical Damping Calculator: c_c = 2 sqrt(k m)

The damping at which a vibrating system returns to rest fastest without overshooting.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • c_c = 2\sqrt{km}
  • k = \frac{c_c^2}{4m}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the critical damping coefficient: critical damping coefficient = 2 x spring stiffness^0.5 x mass^0.5
  2. Put the values in (every value in its SI unit): critical damping coefficient = 2 x 5000^0.5 x 20^0.5
  3. The answer: critical damping coefficient: 632.456 N s/m

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Spring stiffness; Critical damping coefficient (leave it as it is if this is what you are finding) 632.456 N s/m; Mass (leave it as it is if this is what you are finding) 20 kg.

  1. The formula, arranged to find the spring stiffness: spring stiffness = ((critical damping coefficient) / (2 x mass^0.5))^2
  2. Put the values in (every value in its SI unit): spring stiffness = ((632.456) / (2 x 20^0.5))^2
  3. The answer: spring stiffness: 5000.01 N/m
  4. The same answer in other units: 5.00001 N/mm

Common questions

What is critical damping?

The smallest damping that stops a system from oscillating. The mass returns to rest as fast as it can without overshooting.

How is it related to the damping ratio?

The damping ratio is the actual damping divided by the critical damping. A ratio of 1 is critical damping.

What is the formula?

Two times the square root of the spring stiffness times the mass.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any vibrations or dynamics text (damped free vibration).

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