Damping Ratio

阻尼比Advanced

A dimensionless number measuring how fast oscillation decays — it determines whether a system overshoots and rings before settling.

The damping ratio ζ describes how heavily damped a second-order system is, such as a mass-spring-damper system: ζ = c / (2√(km)), where m is mass, k is spring stiffness, c is the damping coefficient, and the denominator 2√(km) is called the critical damping. The equation of motion can be written ẍ + 2ζωₙẋ + ωₙ²x = 0, where ωₙ = √(k/m) is the natural frequency. ζ < 1 is underdamped: the system overshoots the target and oscillates back and forth before settling; ζ = 1 is critically damped: no overshoot, and the fastest possible return to equilibrium; ζ > 1 is overdamped: no oscillation, but a slower return. Joint PD control can be viewed as a virtual spring (Kp) plus damping (Kd), so tuning the gains is really tuning stiffness and damping ratio. MuJoCo's contact parameter solref is likewise specified using a time constant and a damping ratio, which is usually set to 1 (critical damping).

ExampleTreat a joint as a rotor with 0.1 kg·m² of inertia and Kp = 40 N·m/rad; critical damping then requires Kd = 2√(40×0.1) = 4 N·m·s/rad. Any smaller Kd and the joint will oscillate as it settles into position.

Also called: ζ, Critical Damping, Underdamped / Overdamped
Related: Damping · Mass-Spring-Damper System · Natural Frequency · Proportional-Derivative Control · Stiffness and Damping Gains · Step Response Metrics (Overshoot / Settling Time / Steady-State Error)
Sources: Wikipedia: Damping（damping ratio）
MuJoCo Documentation: Modeling – Solver parameters（solref: timeconst, dampratio）