Mechanism Actuator Sizing Calculator — Exact Linkage Torque, Dead Points & Pneumatic Cylinder Force (ISO 15552)

As the output link (or bell-crank) rotates, the rod-end swings on an arc while the cap pivot stays fixed on the frame — so the direction of the actuator force changes continuously. The only correct moment arm at a given pose is the perpendicular distance from the drive pivot to the current cap→rod-end line. The calculator recomputes that line, and the arm d_i(θ), at every step of the sweep for each actuator, so the available torque curve is exact rigid-body statics.

The widespread shortcut of multiplying the actuator force by a fixed input-arm radius (F·r_in) silently assumes the force is always perpendicular to the input arm. Near the ends of travel — exactly where cylinders are weakest — that assumption fails badly: on the mechanism this engine was validated against, the shortcut overstated the single-actuator safety factor by roughly 2.9×. The kinematic chain itself was validated against a physical Autodesk Inventor assembly (a pipe-diverter valve) to a maximum error of 0.04° at 4 measured poses.

In the slot-crank topology the actuators drive a bell-crank about its own pivot; the crank carries a slot, and a pin on the output link rides that slot. The coupling transmits force along the slot's contact normal, so the torque reaching the output link is the crank torque multiplied by the coupling's mechanical advantage — the ratio of the two perpendicular distances from the output pivot and the crank pivot to that shared normal. The crank angle for each output angle is solved by warm-started (continuation) Newton iteration, which keeps the solver on the physically correct branch through the whole sweep.

A dead point is any pose where the drive path collapses — every actuator's moment arm (lever topology) or the slot-coupling arm (slot-crank topology) falls below the validated 8 mm threshold. At a dead point no amount of actuator force produces usable torque, so the tool reports it as a hard failure. The sweep also tracks how far the pin travels along the slot: utilisation above 90% of the slot half-length is warned (little margin for tolerance stack-up) and over-travel beyond the slot fails the check. Optionally, the peak pin force is checked as an ISO 4379 projected bearing pressure p = F_pin/(d_o·t) against a bronze-bush allowable (default 40 MPa).

For pneumatic actuators the axial force is the working pressure times the ISO 15552 piston area — the full bore on the push stroke, and the rod-side annulus on the pull stroke. Because the annulus is always smaller, a reversing mechanism is governed by the weaker pull stroke; the default auto-worst mode conservatively sizes against the weaker of the two, and reports which stroke governed. Note p·A is the catalog thrust: regulator droop, seal friction and meter-out back-pressure typically deliver only ~0.7× of it in service, so the calculator accepts a delivery derate η (0.3–1) and warns whenever the undamped catalog value is used.

The resisting torque is built as a constant worst-case bound from four components: a constant kinetic friction torque about the output pivot; an inertia term I·α, where the angular acceleration comes from a triangular velocity profile over the specified move time; a gravity term for vertical-plane swings that holds the centre-of-gravity arm horizontal over the whole sweep (sin γ ≤ 1 at every instant, so it is a proven upper bound); and a constant process torque for seals, product or wind. Holding this bound constant over the sweep is deliberately conservative — the true resisting torque can only be lower.

The safety factor at each angle is the available torque divided by that bound, evaluated for each actuator alone and for all together. A minimum SF below 1 anywhere in the range means the mechanism cannot be driven; below 2 is flagged as thin for a compressible pneumatic drive with no reserve for stiction, supply droop or load variation. This load model is an engineering estimate — it is surfaced as such in the tool — while the kinematics and torque geometry feeding it are exact.