Beta. These calculators are in beta. We work to make them right — every formula is written out under “Show the math” and the results are checked against published catalogue examples — but we take no responsibility for mistakes or for decisions made on them. Check the math, and check your supplier's data, before you order anything. Found a problem with a calculation? Write to [email protected] — say which calculator and what you entered.

How do I calculate the torque of a slider-crank mechanism?

The crank fixes the motion: from the crank angle follow the slider's position, velocity and acceleration in closed form. The force the crank must put on the slider is the inertia force plus whatever resists the motion, and that force becomes crank torque through the transmission factor sin(φ+β)/cos β — zero at the dead centres, one at mid-stroke. Sampling one revolution gives the torque swing, its peak and its RMS, ready for the motor check.

What you need

crank radius, rod length, crank speed
slider mass, the resisting force and guide friction
gearbox ratio and efficiency, optionally the inertias for the ratio

The math: Slider-crank — motion and torque over one revolutionopen — every formula, so anyone can check it

Crank radius R, rod length L, γ = R/L, crank angle φ (0 = outer dead centre), constant speed ω:

sin β = γ·sin φ                              rod angle
x  =  R·cos φ + L·cos β                       slider position from the crank axis
v  = −R·ω · sin(φ+β) / cos β
a  = −R·ω² · [ cos φ + γ·(cos 2φ + γ²·sin⁴φ) / cos³β ]

F = m·a + sign(v)·(F_ext + μ·m·g)             force the crank must put on the slider
T_crank = F · dx/dφ = −F · R · sin(φ+β)/cos β   sin(φ+β)/cos β is the transmission factor
T_motor = T_crank / (η·i) when driving,  T_crank·η / i when the slider drives the crank

At the dead centres the transmission factor is zero, at 90° it is one, and slightly above one just before. Over a revolution the slider covers two strokes, so its mean speed is 4·R·n/60 whatever the rod. At constant speed the crank's and rotor's inertia add no torque — they are what keeps the speed constant against the swing; they enter only the inertia ratio.

Checked against finite differences of the position, the two-strokes-per-revolution rule, the long-rod (simple harmonic) limit and F = m·a at mid-stroke.

Package mech-crank (calcpkg/1, open format) — pure functions with tests; the browser version is held against it number for number.

The math: Motor check — can this motor make this move?open — every formula, so anyone can check it

T_rms = √( Σ T_i²·t_i / (Σ t_i + t_dwell) )        heating torque over the cycle
duty  = Σ t_i / (Σ t_i + t_dwell)
ratio = J_load_reflected / J_rotor

servo    T_peak ≤ T_peak,motor      T_rms ≤ T_rated      n ≤ n_rated (up to n_max: warning, torque derates)
         inertia ratio ≤ 10 fine, 10–30 warns, > 30 fails (gear it down by i ≈ √(ratio/10))

stepper  f = n/60 · 360/step angle                 full steps per second
         T_peak · safety factor (2) ≤ pull-out torque at f (linear interpolation of the curve)
         an unramped move must start with f ≤ f_self-start
         inertia ratio ≤ 5 direct, ≤ 10 geared (warn);  duty > 50 % warns;  n > 1000 rpm: consider closed loop

dc       n_motor = n·i,   T_motor = T / (i·η)
         T_available = T_stall · (1 − n_motor/n0)      the straight torque–speed line
         T_motor · 1.3 ≤ T_available,   n_motor < n0,   T_rms,motor ≤ ½·T_stall (thermal)
         short of torque → i ≈ n0 / (2·n) puts the motor at its maximum-power point

Checked against Oriental Motor's AZM66AC selection example: required torque with the safety factor and the inertia ratio reproduced within 1 %.

Package motor-move-check (calcpkg/1, open format) — pure functions with tests; the browser version is held against it number for number.

Questions people ask

Why is the torque zero at the dead centres?

At the ends of the stroke the rod is in line with the crank: any slider force passes straight through the crank pin into the bearing and makes no torque. That is why a crank needs a flywheel to get through them.

What does the rod ratio γ = R/L do?

A short rod (large γ) makes the motion asymmetric — faster and harder-accelerating near the outer dead centre — and lifts the peak acceleration above R·ω². As L grows the motion tends to a pure sine.

What if my crank stops between strokes?

This calculator assumes constant crank speed. For a crank that indexes, add the ramp torque of the rotary-table calculator on the crank angle.