The top picture has the connection to the beam (cyan) at half way position, and the correcting link (blue) equal to half the beam length, The right hand end of the beam is constrained to move in a straight horizontal line. The left end of the beam where the piston rod connects moves in a dead straight line. In the second picture I've moved the connection to the beam to 2/3 the way along the beam, and made the correcting link 1/3 of the beam length. The angle of the correctig link to horizontal is now 31 degrees, and the left hand end of the main beam now moves in a shallow arc. With the beam going 15 degrees up and down, the error is 0.1103. If the beam were just anchored at its right hand end, the error would be 3(1-cos(15))=0.1022, so this is not a good linkage. The third picture shows a linkage with the anchor point for the left hand end of the correcting link moved off the piston centre line. The text book relationship is B=(L-A)^2/A. If I plug in L=3, A=1.4 I get B=1.829, and the right hand end of the beam moves in a very shallow arc, the error being 0.0002 according to the CAD, which isn't bad.
By playing tunes with the link lengths, you can get it to generate a long radius arc, which could be of use in some applications.
This linkage is attributed to Scott Russell. I'm not sure whether this is the same chap who constructed the Great Eastern. It forms the basis of Joy's valve gear
Edited By duncan webster on 16/10/2022 19:29:54