On the matter of Lead:
Every text I have seen (including ones written for professional use) states that with notching up, the lead with –
– Stephenson's Link Motion ( and presumably its similar ones like Gooch and Allen) increases;
– Radial Gears (Hackworth and its equivalents, Walschaert's / Baker, etc). remains constant.
In the simpler radial gears the die block locus is an ellipse whose horizontal (valve-spindle aligned) axis in mid-gear is the [2(lap + lead)] part. The ellipse is tilted to increase the component to add the port opening. It is somewhat distorted by having to limit the expansion-link length. Walschaearts and the similar Baker Gears use the combination-lever to give the lap and lead travel.
The link motions give the lap, lead and port opening as a direct resultant of the eccentrics' throws.
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Some simple narrow-gauge locos use Hackworh Gear, but this is heavily perturbed by the chassis suspension, minimised only by very stiff driving-axle springs. However, these machines ambled around within industrial sites where efficiency could be compromised a bit for simplicity and economy of manufacture, use and servicing. The more sophisticated link and radial gears used on Standard Gauge locomotives designed for best efficiency and performance practicable, are better at handling the modest axle suspension travels.
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I don't know its name but the gear on Luker's video image appears a combination of Stephenson's and Gooch; the reverser moving the valve radius-link (G); but the expansion link curve faces the eccentrics (S) from a fixed suspension (also G). The animation shows the gear in one position so I could not determine if this is so.
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JA points to the huge number of valve-gears invented to control variably, reciprocating valve travels and phases. All doing the same thing! I imagine some were as much patent avoiding as trying to make a better valve-gear./ Why for example Baker gear? It is Walschaerts with relatively minor difference, as far as I can see. Marshall, Bremme and Case are all variations of Hackworth's.
Perhaps the most original was the Heywood gear, designed to squeeze under the low running-plates on Heywood's very narrow-gauge estate-railway locomotives. It uses a set of pivoted links in place of the expansion-link.
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Yes, if you want to write a computer-simulation of a gear like Walschaerts' you need use a spread-sheet to handle far more cut-off settings than any real reverser-stand will give; but how did M. Egide Walschaert himself manage?
Slide-rule and log.trig. tables drawn and primed, I think he would have calculated or drawn on paper, the full, mid-gear and a couple of intermediate cut-offs at dead-centre and mid-stroke. If these are all consistent, the intervening points should work properly. They are quite long-winded calculations, with a lot of trigonometry. Though at least the equations were direct, not wrapped in 'Excel'-ese!
(I used 'Excel' frequently, at work…. Chasing formulae brackets I'd miscounted, while cursing its writers for not knowing 0º = 360º or giving proper label editing tools, on polar graphs they called "radar charts" ! )
The simplest and theoretically most efficient eccentric-driven, reversing valve-gear may be the Mann Gear, in which the eccentric is slotted to move across a square on the crankshaft, by a cam arrangement. As far as I know It was only ever used on the Mann steam-wagons; never on locomotives.