Rotational Dynamics

Formula

Description

Newton's second law for rotation: torque equals moment of inertia times angular acceleration. This is the rotational analog of F = ma. The moment of inertia J represents how much torque is needed to achieve a given angular acceleration, analogous to how mass represents resistance to linear acceleration. Higher inertia loads require more torque (and therefore more motor current) to accelerate and decelerate. In servo systems, the motor must provide enough torque to overcome both the load torque and the inertial torque during acceleration phases.

Variables

• τ — Net torque (N·m)
• J — Moment of inertia (kg·m²)
• α — Angular acceleration (rad/s²)

Practical Notes

For optimal servo performance, match the motor inertia to the load inertia (Jload/Jmotor ratio of 1:1 to 5:1). Through a gear reducer, the reflected inertia is Jreflected = Jload/N². Time to reach target speed: t = ω/α = J×ω/τ. Common inertia formulas: solid cylinder J = 0.5×m×r², thin ring J = m×r², solid sphere J = 0.4×m×r².