Is Chebyshev linkage an inversion of four bar linkage?

Answer 1

Well, yes: but that isn't very interesting. The Chebyshev linkage does have four bars, and by definition, every four bar linkage has four inversions.

An 'inversion' of a linkage just means you're holding a specific bar still, or in kinematic terms, you're using that link as the 'ground link'. If a linkage has arms A, B, C, and D, the first inversion uses A as the ground link, the second inversion uses B as the ground link, the third inversion uses C as the ground link, and the fourth inversion uses D as the ground link.

Inversions matter because of the 'coupler' link, which is the one opposite the ground link. Take the first inversion of A, B, C, D above: A is the ground link, so it doesn't move relative to the ground. B and D connect to A, so both of them have one end that doesn't move relative to the ground. The other ends of B and D move in circular arcs relative to the ground (they're lines of constant length, with one end fixed relative to the ground.. in other words, a compass). Link C is the coupler for this inversion, because it's connected to the moving ends of both B and D. It moves in and interesting way relative to the ground.

For each inversion, a different link is the coupler, and the motion of the coupler relative to the ground tends to be VERY different from one inversion to the next. The motion of any given point on the coupler relative to ground is a sixth-order curve (meaning it start with x to the 6th power and works down from there), and people are still doing hard work on the math.

The Chebyshev linkage is interesting because the center of the coupler moves in almost exactly a straight line. It wobbles a little, but the wobble is small enough that they could ignore it when making parts for steam engines that needed to move in a straight line.

An interesting side effect of coupler curves being sixth-order functions is that there are three completely different linkages whose couplers will move the same way. Chebyshev was one of the people who did the math to prove that. The three linkages whose couplers move the same way are called 'cognates' of each other.

The Chebyshev linkage is symmetric, so both of its cognates look the same. They're called the 'Hoekens' linkage, after the guy who did that bit of math.