Machine Kinematics
solved, then drawn
Concept

Kinematic inversion — where it appears

The choice of which link of a chain is bolted to the bench, made after the chain itself is fixed. A chain of four links gives four mechanisms, whose relative motions are identical and whose behaviours are not — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

One chain, four mechanisms. The same four bars and the same four pins in every panel. What changes is which link is bolted to the bench, and that is not a property of the chain — it is a decision about where the bench is. The four mechanisms are crank rocker, double crank, crank rocker, double rocker: one input turns fully in some and rocks in others, and what each one is for is different. What cannot change is the shape of the closed loop, and the two diagonals measure that without reference to which link is held still: swept independently, all four visit the same locus of diagonal pairs to within 2.9e-3, half the sampling resolution. This is why the Whitworth quick-return and the oscillating-cylinder engine are not merely similar to a slider-crank; they are one.

One chain, four mechanisms

Which link of a four-bar is bolted to the bench is not a property of the chain. It is a decision about where the bench is, and making a different one gives a mechanism that looks and behaves completely differently while being, as a chain, the same object — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.

linkages · Fourbar The chain's whole configuration space, drawn on its two angles. A slider-crank asks one thing of its two angles: a cos θ + b cos φ = e, with θ the crank's angle and φ the rod's. Each panel is the square of those two angles from −π to π, with the curve that equation cuts out, followed by arclength on the equation alone. The number under each panel is how many whole turns θ, φ and φ − θ make around a circuit. A curve that crosses the square from side to side carries a turn in that angle; a closed loop inside the square carries none. The two regions where a member turns have two circuits each and the two where none does have one, which is what the four-bar regions these came from predict.

Three rotations, and four benches

The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.

linkages · Fourbar The body points whose three images are in a line. The three prescribed poses, faint, and a curve through them. At every point of a 110×110 grid over the moving body, the point is placed in all three poses and the signed height of its image triangle is measured; the curve is where that length is zero. Those are the points whose three images lie on a line, so the dyad they want is a slide rather than a crank. The curve is a circle. Refined onto the contour, its points lie on the circle through the three image poles to 2e-15, where the pole triangle's own circle misses them by up to 1.98. Its radius is 8.767, so inside a window three units across it reads as a gentle arc; 2 of the three image poles are in the frame and the third, P₁₃, is 6.8 units away.

Where a pin becomes a slide

Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.

synthesis · Synthesis Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images.

The frame seen from the coupler

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

curvature · Curvature

Named alongside it

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

Grashof's conditionSlider-crankInflection circleInstant centreKinematic chainQuick-returnRelative motionBurmester theoryCentrodeCircuitCircumcentreClassification

All concepts