Gresley conjugated valve gear

The Gresley conjugated valve gear is a valve gear for steam locomotives designed by Sir Nigel Gresley, chief mechanical engineer of the LNER, assisted by Harold Holcroft. It enables a three-cylinder locomotive to operate with only the two sets of valve gear for the outside cylinders, and derives the valve motion for the inside cylinder from them by means of levers (the "2 to 1 lever" and "equal lever"). The gear is sometimes known as the Gresley-Holcroft gear, acknowledging Holcroft's major contributions to its development.

The Gresley conjugated gear is effectively an adding machine, where the position of the valve for the inside cylinder is the sum of the positions of the two outside cylinders, but reversed in direction. It can also be thought of as a rocking lever between one outside cylinder and the inside cylinder, as is common on 4-cylinder steam locomotives, but with the pivot point being moved back and forth by a lever from the other outside cylinder.

If we approximate the motion of each valve by a sine wave — if we say the position of a valve in its back-and-forth travel is exactly proportional to the sine of the "driver angle", once we have set the zero point of driver angle at the position it needs to be for that valve — then the mathematics is simple. The position of the valve that is pinned to the long end of the 2-to-1 lever is $\scriptstyle \sin \;\theta$ , while the positions of the other two valves are supposed to be $\scriptstyle \sin(\theta +120^{\circ })$ and $\scriptstyle \sin(\theta -120^{\circ })$ . The position of the short end of the 2-to-1 lever is $\scriptstyle -{\tfrac {1}{2}}\sin \theta$ —which, it turns out, is midway between $\scriptstyle \sin(\theta +120^{\circ })$ and $\scriptstyle \sin(\theta -120^{\circ })$ for any value of $\scriptstyle \;\!\theta$ . So a 1-to-1 lever pivoted on the short arm of the 2-to-1 lever will do the trick.

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