What is the damping ratio?

The damping ratio (zeta) is a dimensionless parameter that describes how oscillations in a second-order system decay after a disturbance. A value less than 1 means the system oscillates (underdamped); exactly 1 gives the fastest non-oscillatory response (critically damped); greater than 1 gives a sluggish, non-oscillatory response (overdamped).

What values of M, C, and K do I enter?

M is the equivalent mass (kg), C is the viscous damping coefficient (N·s/m), and K is the spring stiffness (N/m). For electrical analogs, use inductance, resistance, and 1/capacitance in consistent units. The tool works for any consistent unit system.

What is percent overshoot?

Percent overshoot (PO) is how much the step response exceeds the final steady-state value, expressed as a percentage. It only applies to underdamped systems (zeta < 1) and equals exp(-pi * zeta / sqrt(1 - zeta^2)) * 100.

How is settling time estimated?

Settling time to within 2% of the final value is approximated as 4 / (zeta * omega_n) for underdamped systems. This is the "2% criterion" envelope formula widely used in control design.

Can I use this for electrical RLC circuits?

Yes. Map L (inductance) to M, R (resistance) to C, and 1/C_cap (reciprocal of capacitance) to K. The damping ratio formula is identical; only the physical units differ.

What does critically damped mean in practice?

A critically damped system (zeta = 1) returns to steady state as fast as possible without any overshoot. It is the ideal target for many servo and positioning systems where speed and precision both matter.

Damping Ratio Control

Name: Damping Ratio Control
Availability: InStock
Author: Nham Vu

Enter the system parameters of a second-order control system to instantly calculate the damping ratio, natural frequency, and response classification.

System Parameters

Enter mass (M), damping coefficient (C), and stiffness (K).

Mass M (kg)

Damping Coefficient C (N·s/m)

Stiffness K (N/m)

Quick Presets

Enter parameters and click Calculate

System Classification

Damping Ratio ζ

Natural Freq. ω_n

rad/s

Critical Damping C_cr

N·s/m

Characteristic Roots

Standard Form Transfer Function

G(s) = ω_n² / (s² + 2ζω_ns + ω_n²)

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Summary