Natural Frequency

The frequency at which a system oscillates on its own with no sustained outside force; matching this frequency causes resonance.

Natural frequency is the frequency at which a vibrating system oscillates on its own, with no sustained external excitation. For the simplest mass-spring system, ω₀ = √(k/m), where k is the spring stiffness and m is the mass, in rad/s (divide by 2π to convert to hertz). The stiffer the spring or the lighter the mass, the faster it vibrates. When an external force's frequency approaches the natural frequency, the amplitude grows sharply — this is resonance. It matters for robots in two ways. One is structural: harmonic drives, long slender links, and flexible joints all have some elasticity, so the whole machine has natural frequencies, and either too-high control gains or a sudden change in trajectory acceleration can excite unwanted vibration. The other is control: under PD control, a joint's error obeys a second-order equation whose natural frequency is ωn = √(Kp/M) (Kp is the proportional gain, M is the rotational inertia), which, together with the damping ratio, determines how fast the response is and how much it overshoots.

ExampleWith a joint inertia M = 0.1 kg·m² and proportional gain Kp = 10 N·m/rad, ωn = √(10/0.1) = 10 rad/s, about 1.6 Hz; raising Kp to 40 doubles ωn to 20 rad/s, giving a faster response but also making it easier to excite structural vibration from unmodeled flexibility.

Also called

ωn, Resonant Frequency

Related

Damping Ratio · Mass-Spring-Damper System · Stiffness · Flexible Joint · Proportional-Derivative Control · Control Bandwidth

Sources

Wikipedia: Natural frequency
Lynch & Park, Modern Robotics（2017）11.2–11.4 节：二阶误差动力学与 PD 增益 (Chinese)