Machine Kinematics
solved, then drawn
Concept

Chordal action — where it appears

The variation in a chain drive's effective radius within every tooth, because the pins sit on a polygon rather than on the pitch circle. It is 1 − cos(π/z) of the pitch radius, which is four per cent on an eleven-tooth sprocket and falls away as the square of the tooth count.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

A chain is a strand with a smallest piece. 53 teeth driving 11 at a pitch of 12.7 mm. The taut strand is drawn as this field computes it; inside each pitch circle is the polygon the pins actually sit on. The two agree on the straight spans and not on the wraps, and the gap is what the applied field's chordal action is: the effective radius swings between R cos(π/n) and R within every tooth, which is 4.05% on the small sprocket and 0.176% on the large one. A strand has no such number, because a strand has no pitch. positioned by solving, not by drawing.

A chain is not a strand

A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.

strands · Strand 53/11, instant by instant. The ratio a 53-tooth sprocket and a 11-tooth sprocket actually deliver, against the driver's rotation. The quoted 4.818 is the flat line, and it is exactly right as an average — the mean over a turn comes out at 4.8182, which the computation is never told. Within every tooth the ratio swings from 4.756 to 4.948, a fluctuation of 3.99%. The sharp corners are pins seating: the identity of the pin the strand runs from changes 53 times a turn on one sprocket and 11 times on the other, and the two do not coincide.

The chain is a polygon

A chain's pins sit on a polygon, so the radius that matters swings by 1 − cos(π/n) within every tooth: four per cent on an eleven-tooth sprocket. The quoted 53/11 is the mean of that, exactly — and how much of the fluctuation reaches the back wheel is decided by the fractional number of pitches in the taut strand, which nobody adjusts and which can take the drive from perfectly uniform to two per cent.

applied · Tooth A cycloidal rise on a chain-driven camshaft. A cycloidal rise over 120° of cam angle, the camshaft driven by a roller chain moving at steady speed over an 11-tooth sprocket. The chain leaves the sprocket at a radius that varies as each pin seats, so the shaft speeds up and slows once per tooth, and its angular acceleration jumps from one sign to the other at every seating. The follower's acceleration (solid) therefore steps at every tooth — 4 or so inside this rise — by up to 0.414 of the steady peak, and the cycloidal law's continuous acceleration, the reason it is chosen, is lost. At the worst clocking against the teeth the peak is 1.165 times the steady shaft's (dashed). Dragging changes the tooth count.

A cam on a chain steps at every tooth

A roller chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so a chain-driven camshaft speeds up through every tooth and its angular acceleration jumps back at every seating. That puts a step into the follower's acceleration once per tooth, whatever the law — a cycloidal rise, chosen for its continuous acceleration, gets four steps inside a 120° rise on an eleven-tooth sprocket, each 0.41 of its peak. The harm falls with the tooth count, as one over it, not up as its square, and the simple harmonic law, whose peak sits where its velocity is nought, is hurt least of all.

cams · Cam

Named alongside it

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

ApproximationChain drivePitchSprocketAccelerationCentre distancecycloidal motionDerivativeDesign ruleFollowerJerkMesh

All concepts