Machine Kinematics
solved, then drawn
Concept

the Hooke joint — where it appears

The cross-and-two-yokes coupling between two shafts that meet at an angle, the same object as a universal joint. Its output turns fast and slow twice a turn, between the cosine of the bend and its reciprocal, so anything driven through one sees an uneven shaft.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A universal joint at 40° input, shafts 25° apart. Two shafts meeting at 25°, joined by a cross whose two pins are at right angles to each other and each at right angles to the shaft it carries. That is the whole geometry, and everything else follows from it. This is a four-joint spatial loop with all four axes through one point: Kutzbach counts −2 and the screw system has rank 3, so it has one degree of freedom and turns. At this instant the output shaft is at 42.79° while the input is at 40°, and the output is turning 1.0124 times as fast — which is not 1, and never is except at the four points of each turn where the curves cross.

The joint that is not constant velocity

A universal joint is the spatial mechanism everybody has met and almost nobody has been told the truth about. Its output shaft runs fast, then slow, twice per revolution, and the amount depends only on the angle between the shafts — which is why cars have two of them and why the second one has to be fitted the right way round.

spatial · Spatial The same cycloidal rise behind a bent joint. A cycloidal rise over 120° of cam angle, with the camshaft driven at constant speed through a Hooke's joint bent at 30°. Across: the fraction of the rise; up: the follower's acceleration in time, divided by its peak on a steady shaft. The joint turns the camshaft unevenly with a period of half a turn, so where the rise falls in that period — the clocking, the cam angle at which the rise begins, measured from where the shaft turns fastest — decides what the follower feels. At the worst clocking, 100°, the peak is 1.352 times the steady one; at the best, 30°, it is 0.828 — lower than on a steady shaft. Every curve still starts and ends at nought. Dragging changes the bend.

A cam behind a bent joint

A cam's motion law is written in cam angle and delivered in time, and the two are the same only on a shaft that turns steadily. Drive the camshaft through a Hooke's joint and the follower's acceleration gains a second term — the law's velocity times the shaft's angular acceleration — that a steady shaft does not have. Where the rise falls in the joint's half-turn then decides everything: at a 30° bend the same cycloidal rise peaks 35% above its steady value or 17% below it. And past a bend of 34.5° a well-clocked cycloidal cam gives its follower less peak acceleration than the constant-acceleration law, the one designed to minimise it.

cams · Cam

Named alongside it

The objects these essays reach for when they reach for this one.

Universal jointAccelerationConstant velocitycycloidal motionDwellFollowerGrashof's conditionJerkMotion lawRatioScrewSpherical linkage

All concepts