Paradoxical mechanism

Muse­ums and archives

Mech­a­nism is stored in the Poly­tech­nic Museum (Moscow, Rus­sia); fund repos­i­tory, PM № 19461.

Mech­a­nism is stored in the Musée des arts et métiers du Con­ser­va­toire national des arts et métiers (Paris, France); CNAM № 11472-0007.

Mech­a­nism is stored in the Museum of the His­tory of Physics and Math­e­mat­ics of Saint Peters­burg State Uni­ver­sity (Peter­hof, Rus­sia).


Research

I. I. Arto­bolevsky, N. I. Lev­it­sky. Tcheby­shev's Mech­a­nisms / In: P. L. Tcheby­shev’s Sci­en­tific Her­itage. Iss. 2. The­ory of mech­a­nisms. — Moscow-Leningrad: AS USSR. 1945. P. 30–32. (Russ­ian)

I. I. Arto­bolevsky, N. I. Lev­it­sky. Mod­els of mech­a­nisms of P. L. Tcheby­shev / In: The com­plete works of P. L. Tcheby­shev. Vol. IV. The­ory of mech­a­nisms. — Moscow-Leningrad: AS USSR. 1948. P. 215–217. (Russ­ian)


Descrip­tion

Which trans­for­ma­tion of curves can per­form the given link­age with a sin­gle fixed red hinge?

Let the gray hinge fol­low the curve sym­met­ric with respect to the line pass­ing through the fixed red hinge. One can show that in this case the tra­jec­tory of the blue hinge will also be sym­met­ric with respect to some line pass­ing through the fixed hinge. A russ­ian math­e­mati­cian Pafnuty Lvovich Tcheby­shev inves­ti­gated the ques­tion of this tra­jec­tory.

An impor­tant case of the gray tra­jec­tory is a cir­cle.In prac­tice it's real­ized by adding a fixed (red) hinge and a lead­ing link of some length.

Impor­tant cases of the blue tra­jec­tory are those sim­i­lar to a line seg­ment, a cir­cle or its arc. Tcheby­shev writes: "Here we con­sider the cases that are the sim­plest and most com­mon in prac­tice, that is to obtain motion along a curve, a sig­nif­i­cant part of whom doesn't dif­fer much from a cir­cle arc or a straight line."

Exactly to the ques­tion of find­ing the best para­me­ters of this mech­a­nism, solv­ing prac­ti­cal prob­lems, Pafnuty Lvovich applies for the first time the the­ory of func­tion approx­i­ma­tion, that he devel­oped study­ing the Watt par­al­lel­o­gram.

Choos­ing the dis­tance between the fixed hinges, the length of the lead­ing link and the angle between the links, Pafnuty Lvovich obtains a closed tra­jec­tory not dif­fer­ing much from a straight-line seg­ment. The devi­a­tion of the blue tra­jec­tory from a straight one can be decreased chang­ing the para­me­ters of the mech­a­nism. How­ever, the stroke of the blue hinge will decrease as well. But this hap­pens more slowly that the decrease of the devi­a­tion, so one can choose appro­pri­ate para­me­ters for a given appli­ca­tion. This is one of the vari­ants of an approx­i­mate straight­en­ing mech­a­nism pro­posed by Tcheby­shev.

Let's move to the case of sim­i­lar­ity of the blue curve with a cir­cle.

Con­sid­er­ing the case when the links form a line, we come up with a mech­a­nism that looks like the greek let­ter lambda. Tcheby­shev used it with some para­me­ters to build the first «tan­gent to two con­cen­tric cir­cles, remain­ing between them. By chang­ing the para­me­ters of the mech­a­nism, one can reduce the dis­tance between the con­cen­tric cir­cles, within which is the blue tra­jec­tory.

Improve the lambda-mech­a­nism by adding a fixed hinge and two links, whose sum of the lengths is equal to the radius of the larger cir­cle, and the dif­fer­ence: the radius of the smaller one.

The mech­a­nism we get has bifur­ca­tion points, or as one says, sin­gu­lar points. Being in such a point dur­ing the same clock­wise motion of the lambda-mech­a­nism, the links may six­such bifur­ca­tion points: when the added links are on the same line.

There is a big and impor­tant brach of math­e­mat­ics, sin­gu­lar­ity the­ory, that stud­ies sin­gu­lar points. A very sim­ple and impor­tant case is the study of func­tion behav­ior through study­ing its points of max­i­mum and min­i­mum.

In order for our mech­a­nism to pass the six sin­gu­lar points in a pre­s­e­lected direc­tion, a small link is asso­ci­ated with a fly­wheel, which being pro­moted in some way, con­ducts the mech­a­nism from a sin­gu­lar point rotat­ing in the same direc­tion.

If one twists the fly­wheel, as well as the lead­ing link, clock­wise from the point of bifur­ca­tion, then in one turn of the lead­ing link the fly­wheel will do two turns.

If one twists the fly­wheel, coun­ter­clock­wise from the point of bifur­ca­tion, then in one clock­wise turn of the lead­ing link the fly­wheel will do four turns!

Therein lies the para­dox of this mech­a­nism, invented and made by Pafnuty Lvovich Tcheby­shev. It might seem that a flat hinge mech­a­nism must oper­ate unam­bigu­ously, how­ever, as we see, this is not always the case. And the rea­son is the sin­gu­lar point.


All Mechanisms

Reconstruction
Reconstruction
Reconstruction
Kinematic scheme
Model by Tchebyshev (Polytechnical museum)
Model by Tchebyshev (CNAM)
Model by Tchebyshev (Museum of St. Petersburg University)
Model by Tchebyshev (Museum of St. Petersburg University)
Название