4. CHEBYSHEV LINKAGE


The Inventor
Pafnuty Lvovich Chebyshev (or Tchebicheff) lived from 1821 until 1894. He was a Russian mathematician who founded of the St. Petersburg Mathematical School, and is remembered for his work on prime numbers and probability. He studied theoretical kinematics, particularly rectilinear motion. Chebyshev was a prolific mathematician. In number theory, he worked on the distribution of prime numbers. He then turned to applied mathematics and began a lengthy study of mechanical systems. Besides inventing a number of mechanical linkages, he also developed techniques of numerical analysis for dealing with theoretical calculations in mechanics. He later tackled the fields of probability and statistics, proving important basic theorems including the law of large numbers. Judging by his published works and his reputation abroad, his interest in kinematics and linkages amounted to an obsession. In 1853, after visiting France and England and observing carefully the progress of applied mechanics in those countries, he wrote his first paper on approximate straightline linkages, and over the next 30 years he attacked the problem with new vigor at least a dozen times. Chebyshev is considered a founding father of Russian mathematics. According to the Mathematics Genealogy Project, Chebyshev has about 5,000 mathematical ‘descendants’. The Linkage The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straightline motion. In the figure at right, A, B are fixed pivots. C and D are moveable pivot points, and P is the midpoint of CD. Rotation of B through a circular arc about A results in an approximate linear movement by P. The proportions used in this linkage are AD : BC : AB (= CD) = 2 : 1 : 2.5. This relationship assures that the links AB, CD lie vertically when they are at the extremes of their movement. Then we are given that x : y : z = 2.5 : 1 : 2. Now by placing the linkage in one of its extreme positions with BC, DC vertical and overlapping, we easily obtain


