Force, Torque, Speed, and Power
Actuator sizing starts with force, torque, speed, and power. A motor that has enough torque at zero speed may fail at operating speed. A motor that has enough speed may overheat if it cannot deliver continuous torque. This lesson links mechanical requirements to electrical drive choices.
Learning Objectives
By the end of this lesson, you should be able to:
- convert between force and torque using radius or leadscrew pitch;
- calculate linear and rotational mechanical power;
- convert RPM to angular velocity;
- include efficiency, acceleration, duty cycle, and thermal margin;
- read torque-speed requirements without confusing stall, peak, and continuous ratings.
Force to Torque
Torque is force applied at a radius:
$$
\tau = Fr
$$
where tau is torque in N m, F is tangential force in N, and r is radius in m.
For a belt pulley with radius 20 mm pulling 50 N:
$$
\tau = 50 \times 0.020 = 1.0\ N m
$$
For a leadscrew, the ideal torque estimate is:
$$
\tau = \frac{F p}{2\pi\eta}
$$
where p is screw lead in m/rev and eta is mechanical efficiency. Sliding screws may have low efficiency; ball screws are usually much higher.
Speed and Power
Linear mechanical power:
$$
P = Fv
$$
Rotational mechanical power:
$$
P = \tau\omega
$$
Convert revolutions per minute to radians per second:
$$
\omega = \frac{2\pi \times RPM}{60}
$$
Power does not replace torque. Torque tells whether the actuator can produce effort. Power tells whether it can produce that effort at the required speed.
Worked Example: Wheel Drive
A wheel needs 1.2 N m at 300 RPM.
$$
\omega = \frac{2\pi \times 300}{60} = 31.4\ rad/s
$$
$$
P_\text{load} = 1.2 \times 31.4 = 37.7\ W
$$
If drivetrain efficiency is 70%:
$$
P_\text{motor} = \frac{37.7}{0.70} = 53.9\ W
$$
If the duty cycle is continuous, choose a motor, driver, wiring, and supply that can handle this operating point thermally. If the duty cycle is intermittent, check peak rating duration and cool-down assumptions from the data sheet.
Worked Example: Leadscrew Axis
A vertical axis lifts 80 N using a 5 mm/rev leadscrew with efficiency 0.35.
$$
\tau = \frac{80 \times 0.005}{2\pi \times 0.35} = 0.18\ N m
$$
This is an ideal steady lift torque. Add torque for acceleration, friction variation, coupler losses, and a safe holding strategy. A vertical axis may need a brake or self-locking screw so it does not drop when power is removed.
Torque-Speed Curves
Motors do not deliver maximum torque at every speed.
DC motors have high stall torque, but stall current can destroy windings and drivers. Stepper motors produce useful holding torque but lose torque as speed rises. Servos publish rated and peak operating regions, often with thermal limits.
Electrical Consequences
Mechanical torque becomes motor current. Mechanical speed becomes voltage demand because winding inductance and back EMF limit current rise at speed. That is why a motor may work at low speed on a weak supply and stall at high speed even though the driver current limit is unchanged.
Check:
- continuous current, peak current, and peak duration;
- driver voltage range and supply transient behavior;
- heat sinking and enclosure temperature;
- braking energy, regenerative current, and clamp path;
- wiring voltage drop and connector current rating.
Common Mistakes
- Confusing torque with power.
- Selecting by stall torque without checking operating speed.
- Treating peak torque as continuous torque.
- Ignoring acceleration torque.
- Forgetting gearbox, belt, or screw efficiency.
- Ignoring heat during repeated starts and stops.
Summary
Force and torque create motion; speed determines how fast work happens; power combines effort and speed. Reliable sizing uses worst-case load, acceleration, mechanical efficiency, motor torque-speed curves, driver limits, duty cycle, and thermal margin.
Further Reading
- Maxon, Portescap, and Oriental Motor motor-selection guides.
- Kollmorgen servo motor sizing application notes.
- Machine Design tutorials on torque, speed, power, and duty cycle.