Sorting mechanism

Con­sider a kind of lambda mech­a­nism that trans­lates the uni­form motion of a gray hinge around a cir­cle into the motion of a blue hinge along some curve.

The para­me­ters of the mech­a­nism are cho­sen so that part of the tra­jec­tory of the blue hinge is close to the arc of some cir­cle. The move­ment along this part of the tra­jec­tory takes more than half of the time.

Let's add another link to the mech­a­nism: ”the radius.” As the blue hinge moves along a seg­ment of a tra­jec­tory close to a cir­cle, the free end of the new link will oscil­late about the cen­ter of that cir­cle.

If the free end of ”the radius” con­trols the motion of another link, the yoke, it will remain sta­tion­ary most of the time (in lay­man's terms, geo­met­ri­cally there will be small devi­a­tions), and then make a rapid full roll back and forth. As they say in mechan­ics, it's a mech­a­nism with a long stop of the slave link at the end of its stroke.

This beau­ti­ful geo­met­ric idea was devised by the great­est Russ­ian math­e­mati­cian Pafnuty L. Tcheby­shev to cre­ate an ele­gant mech­a­nism, designed for the impor­tant prac­ti­cal task of sort­ing grain. Both in the 19th cen­tury and today, grain is sorted by weight in order to select the best - heav­ier - grains.

The grain was poured into a box. It has not sur­vived and has been recon­structed from a pho­to­graph. Next, through a curved chan­nel, the grain entered a tray attached to the end of the link, which had a long stop. Since the arm stop lasts more than half the time, dur­ing this inter­val the grain fills the tray. Rapid rolling back and forth will scat­ter the grain. To pre­vent the grain from spilling out of the box at the moment of rolling, to the kine­matic scheme con­sid­ered at the begin­ning of the film. The kine­matic scheme in the mech­a­nism is sup­ple­mented by another chain. ”A pickup,” attached to one of the links, blocks the exit from the box at the moment of rolling.

The sharp ejec­tion from the tray sorts the grains by weight. The best — heav­ier — grains end up (on aver­age) lying fur­ther away from the mech­a­nism.

The bulk of the sort­ing machine is kept in the Museum of the His­tory of St. Peters­burg Uni­ver­sity, as is the photo of the uni­ver­sity's mechan­i­cal cab­i­net (taken by B. N. Men­shutkin in the end of the 19th cen­tury). Based on these sources, this video pre­sents a work­ing exact replica of the mech­a­nism.