Mass-Spring-Damper System

质量-弹簧-阻尼系统Common

The most basic vibrating system, made of a mass, a spring, and a damper, used as an approximation throughout control and contact modeling.

The mass-spring-damper system is the most fundamental dynamic model in mechanics and control: a mass m attached to a spring of stiffness k and a damper with damping coefficient c, with equation of motion mẍ + cẋ + kx = F, where x is displacement from equilibrium and F is an external force. The spring pulls the mass back toward equilibrium, and the damper dissipates energy. It defines a natural frequency ωn = √(k/m) and a damping ratio ζ = c/(2√(km)): ζ < 1 oscillates its way to a stop (underdamped), ζ = 1 returns to equilibrium fastest with no overshoot (critically damped), and ζ > 1 creeps back slowly (overdamped). It's everywhere in robotics: PD control at a joint behaves like a virtual spring plus damper, impedance control makes an end effector behave like a desired mass-spring-damper system, and many simulators compute soft-contact forces using a spring-damper model too.

ExampleIn joint PD control, τ = Kp(q_target − q) − Kd·q̇, Kp acts like spring stiffness k and Kd acts like damping c: too small a Kd and the joint oscillates back and forth, too large and the motion becomes sluggish.

Also called: Spring-Mass-Damper System
Related: Damping Ratio · Natural Frequency · Stiffness · Damping · Impedance Control · Proportional-Derivative Control
Sources: Wikipedia: Mass-spring-damper model