Actuator Selection & Motor/Gearbox Sizing

Choosing the right motor for the job. Learn actuator types (brushed DC, BLDC/PMSM servo, stepper, AC induction, linear motors, piezo, solenoids) and when to use each, move profiles and torque requirements (acceleration, friction, gravity), reflected inertia and gear ratios, inertia matching, RMS torque and thermal limits, speed–torque curve checks, gearbox backlash and efficiency, and a sizing workflow — with Python comparing gear ratios.

An undersized motor overheats or can't follow the motion profile; an oversized one wastes money, space and energy and may even perform worse. Sizing links the mechanical load and motion profile to motor torque, speed and inertia through the transmission.

The process is systematic: define the move, compute load torques and inertia, choose a transmission ratio, check peak and RMS torque and maximum speed against the motor's curves, then verify the driver and power supply.

An analogy

Choosing a gear ratio is like choosing a bicycle gear: low gear (high ratio) makes climbing easy but limits top speed; high gear is fast but hard to start. The right gear lets the rider (motor) work in their comfortable range.

Actuator types

Match the actuator to the task:

Load analysis

Compute all load torques: acceleration (J·α), friction (Coulomb + viscous), gravity (m·g·r·cos θ for arms), process forces (cutting, pressing). Reflect them through the transmission: torques divide by N·η, inertias by N². Linear axes convert via screw lead or pulley radius (J = m·(lead/2π)² or m·r²).

Checks against the motor

Peak torque at each speed must lie inside the motor's intermittent torque–speed curve (with voltage margin); RMS torque must be below the continuous rating (thermal); maximum speed must be below rated speed; inertia ratio ideally below ~5–10:1 for good servo tuning (higher ratios work with stiff couplings and good tuning). Check driver peak/continuous current and supply power, including regeneration during deceleration.

Transmissions and trade-offs

Planetary gearboxes (compact, 1–3 arcmin backlash), harmonic drives (zero backlash, high ratio, robot joints), cycloidal (shock-tolerant), worm (self-locking, low efficiency), belts and screws for linear motion. Direct drive avoids backlash and wear at the cost of larger motors. Optimise over ratio, motor frame and driver with duty-cycle analysis; include temperature rise, bearing loads and life.

Formulas

Worked example: Reflected inertia

Given: J_L = 0.05 kg·m², N = 20

Find: J_refl.

Answer: 1.25 × 10⁻⁴ kg·m² (2.5× the motor)

Worked example: RMS torque

Given: 0.14 N·m for 0.17 s, 0.05 for 0.17 s, 0.1 for 0.17 s, 0 for 0.5 s dwell

Find: T_rms.

Answer: ≈ 0.074 N·m

Worked example: Screw inertia

Given: 20 kg carriage, 10 mm lead

Find: Reflected inertia.

Answer: 5.07 × 10⁻⁵ kg·m²

Comparing gear ratios for a rotating arm

import math
# Motor + gearbox sizing for a rotating arm (trapezoidal move)
J_load = 0.05            # kg*m^2
T_load = 0.8             # N*m gravity/friction at the load
theta = math.pi / 2      # 90 degree move
t_move = 0.5             # s, 1/3 accel, 1/3 constant, 1/3 decel
w_max = theta / (t_move * 2 / 3)          # trapezoid with equal thirds
alpha = w_max / (t_move / 3)
J_motor = 5e-5
for N in (5, 10, 20, 40):
    J_ref = J_load / N**2
    T_acc = (J_motor + J_ref) * alpha * N + T_load / (N * 0.9)
    w_m = w_max * N * 60 / (2 * math.pi)
    print(f"ratio {N:2d}: inertia ratio {J_ref/J_motor:6.1f}, peak motor torque {T_acc:.3f} N*m, motor speed {w_m:.0f} rpm")
print(f"optimal ratio for pure inertia: N* = sqrt(J_load/J_motor) = {math.sqrt(J_load/J_motor):.1f}")
# ratio  5: inertia ratio   40.0, peak motor torque 0.468 N*m, motor speed 225 rpm
# ratio 10: inertia ratio   10.0, peak motor torque 0.244 N*m, motor speed 450 rpm
# ratio 20: inertia ratio    2.5, peak motor torque 0.143 N*m, motor speed 900 rpm
# ratio 40: inertia ratio    0.6, peak motor torque 0.114 N*m, motor speed 1800 rpm

Higher ratios cut the required torque and the inertia mismatch but raise motor speed. Ratio 20 gives a good inertia ratio (2.5:1) and moderate speed; ratio 40 needs slightly less torque but 1800 rpm and more gearbox cost/backlash. The theoretical optimum 31.6 minimises acceleration torque only — practical choices balance speed limits, backlash and cost.

Common mistakes

Applications

Key takeaways

Going further