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Critical Damping Coefficient Calculator

Enter mass and spring stiffness to find the critical damping coefficient.

How it is calculated

For a mass m on a spring of stiffness k, the tool computes three values: the critical damping coefficient cc = 2·√(k·m) in N·s/m, the natural angular frequency ωn = √(k / m) in rad/s, and the natural frequency fn = ωn / (2π) in Hz. Use SI units: mass in kilograms and stiffness in newtons per metre. Results are shown with four decimals.

The mass must be above zero and the stiffness cannot be negative; a stiffness of exactly 0 is accepted and just gives zeros. Nothing here depends on your actual damper: to judge a real system, divide its damping coefficient c by cc, giving the damping ratio ζ = c / cc. Below 1 the motion oscillates, at 1 it is critically damped and above 1 it creeps back without overshoot.

Worked example

A 2 kg mass hangs on a spring of 800 N/m. Product k·m = 800 × 2 = 1600, whose square root is 40, so cc = 2 × 40 = 80.0000 N·s/m. Natural angular frequency: √(800 / 2) = √400 = 20.0000 rad/s. Natural frequency: 20 / (2π) = 20 / 6.2832 = 3.1831 Hz. A damper with c = 40 N·s/m would give ζ = 40 / 80 = 0.5, so that system would still oscillate.

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Common mistakes

  • Entering mass in grams or stiffness in N/mm. The units must be kg and N/m; a spring rated 5 N/mm is 5000 N/m, and skipping that conversion makes cc about 32 times too small.
  • Confusing ωn in rad/s with fn in Hz. They differ by 2π: 20 rad/s is only about 3.18 oscillations per second.
  • Treating cc as the damping your system has. It is the target value for no overshoot, so compare your real damper's c with it to get the damping ratio.
  • Using the weight in newtons instead of the mass in kilograms. A 20 N weight is about 2.04 kg, not 20 kg.
  • Applying it to a system with several masses or springs as if it were one. The tool assumes a single mass on a single spring; combine the springs to an equivalent stiffness first.

Sources

About this calculator

For a mass-spring-damper system, the critical damping coefficient is c_c = 2√(km), where k is the spring stiffness and m is the mass — the exact amount of damping at which the system returns to equilibrium as fast as possible without oscillating. Below this value (underdamped), the system overshoots and oscillates before settling; above it (overdamped), it returns to rest more slowly but without any overshoot. This calculator also reports the natural (undamped) angular frequency ωₙ = √(k/m), which sets the baseline oscillation rate that damping acts against.

Hitting critical damping precisely is the design goal whenever fast settling without overshoot matters more than anything else. Mechanical and automotive engineers apply it when tuning shock absorbers and suspension systems, control systems engineers use it when designing feedback loops (like a servo or thermostat) that must reach setpoint quickly without oscillating, and structural engineers reference it when analyzing vibration dampers on buildings and bridges.

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  • Editorial team

    The result describes the mass and spring only; it says nothing about your damper. To judge a real system, take the damper's coefficient c in N·s/m and divide it by the value shown to get the damping ratio. Door closers and meter movements aim near 1, while car suspensions are usually set well below 1 on purpose, trading a little oscillation for ride feel. Keep mass in kilograms and stiffness in N/m. What system are you tuning?

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