Size a servo motor and gearbox from load dynamics — reflected inertia, a full multi-segment duty cycle for true RMS torque, gravity and friction loads, gearbox inertia, and a live torque–speed operating-point check.
Peak load speed is derived from distance ÷ profile. RMS torque integrates over accel + run + decel + dwell.
Every operating point (accel, run, decel) must sit under the peak curve; continuous-duty points must sit under the continuous curve. Points outside the envelope are drawn in red.
Reflected inertia. Load inertia at the motor shaft is J_L' = J_L / i². Total accelerated inertia is J_M + J_gear + J_L'. Linear masses convert via J = m·r² (belt/rack, r = radius) or J = m·(lead/2π)² (screw).
Acceleration torque. T_acc = J_total · α, where α = ω_peak / t_acc at the motor shaft. S-curve applies a 1.5× peak-jerk factor to α.
Load torque. Friction/process force reflected to the motor as T_f / (i·η). Gravity (vertical) adds m·g·(1−counterbalance) through the drivetrain, aiding on the way down and opposing on the way up — the up-stroke governs sizing.
RMS torque. T_rms = √( Σ Tᵢ²·tᵢ / Σtᵢ ) over accel, run, decel and dwell segments. This is the thermal-equivalent continuous torque the motor must sustain.
Sizing checks. Peak required < motor peak; RMS required < motor continuous; peak speed < max speed; inertia ratio within target band. A safety margin is folded into the utilization read-out.
Torque-speed curves are idealized (flat continuous region to rated speed, linearly falling peak into the field-weakening region). Use the manufacturer's real curve for final selection.