Critical Damping Calculator
Compute the critical damping coefficient cc of a mass-spring system — the threshold value that returns the system to equilibrium as fast as possible without oscillating. 100% runs in your browser, no server.
cc = 2√(k·m) = 2·m·ωnQuick example
A m = 1 kg mass hangs on a spring with stiffness k = 100 N/m. Its critical damping coefficient is cc = 2√(100 × 1) = 2 × 10 = 20 N·s/m. A damper with a coefficient of exactly 20 N·s/m returns the mass to equilibrium as fast as possible without vibrating.
Scope
This tool computes the critical damping coefficient cc itself (in N·s/m). Not the damping ratio ζ = c/cc — computing the damping ratio from an actual coefficient is a separate tool (damping-ratio). This tool also does not model the time response of a damped system; only the critical threshold value and its reference natural frequency.
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Frequently Asked Questions
What is critical damping?
Critical damping is the threshold damping coefficient of a mass-spring-damper system. If the actual damping c equals the critical value c_c, a disturbed system returns to its equilibrium position as fast as possible without oscillating. Damping below c_c makes the system vibrate (underdamped); above c_c the system returns slowly without vibration (overdamped).
What is the critical damping coefficient c_c formula?
The formula is c_c = 2√(k·m), equivalent to c_c = 2·m·ω_n. With m the mass in kg and k the spring stiffness in N/m, the result is in N·s/m. Example: m = 1 kg and k = 100 N/m give c_c = 2√100 = 20 N·s/m.
What is the difference between critical damping c_c and damping ratio ζ?
c_c is the absolute threshold value in N·s/m, while the damping ratio ζ is the dimensionless ratio between actual damping and critical damping: ζ = c/c_c. This calculator computes c_c itself; to compute ζ from a known c, use the separate damping-ratio tool.
What is the unit of the critical damping coefficient?
The unit is N·s/m, identical to kg/s. Proof: 1 N = 1 kg·m/s², so 1 N·s/m = 1 kg/s. This calculator accepts mass in kg, g, or lb, and stiffness in N/m, N/mm, or lbf/in.
How do I compute spring stiffness k from a known c_c?
Invert the formula: k = c_c²/(4m). Example: c_c = 20 N·s/m and m = 1 kg give k = 400/4 = 100 N/m. Mode 2 of this calculator performs that calculation, including output unit conversion.
How do I compute mass m from a known c_c?
The formula is m = c_c²/(4k). Example: c_c = 20 N·s/m and k = 100 N/m give m = 400/400 = 1 kg. Mode 3 of this calculator performs that calculation with kg, g, or lb unit options.
How is c_c related to natural frequency ω_n?
They are linked through c_c = 2·m·ω_n, with ω_n = √(k/m) in rad/s. Example: m = 1 kg and k = 100 N/m give ω_n = 10 rad/s, or f_n = ω_n/(2π) ≈ 1.5915 Hz. Mode 4 shows ω_n, f_n, and its period as reference.
When is critical damping used in practice?
Critical damping is the design target when vibration is unwanted but a slow response is also costly: vehicle suspensions, building seismic dampers, automatic doors, galvanometer needles, and elevator doors. Engineers usually design slightly below or above c_c, depending on whether response speed or the absence of overshoot is prioritized.
Free tool by NRTechStudio