Description
The Chebyshev mechanism was invented in the 19th century by the mathematician Pafnuty Chebyshev, who conducted research theoretical problems kinematic mechanisms. One such problem was the conversion problem. rotary motion in near-rectilinear motion.
Rectilinear movement is determined by the movement of the point P - the midpoint of the link L 3 located in the middle between two extreme points couplings of this four-link mechanism. ( L 1 , L 2 , L 3 , and L 4 are shown in the illustration). When moving along the section shown in the illustration, the point P deviates from the ideal rectilinear movement. The ratios between the lengths of the links are as follows:
Point P lies in the middle of the link L 3 . The given ratios show that the link L 3 is positioned vertically when it is in the extreme positions of its movement.
The lengths are related mathematically as follows:
Based on the described mechanism, Chebyshev manufactured the world's first walking mechanism, which was used great success at the World Exhibition in Paris in 1878.
see also
Other ways to convert rotational motion into approximately rectilinear motion are as follows:
- the Heuken mechanism is a variation of the Chebyshev mechanism;
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An excerpt characterizing the Chebyshev Mechanism
- About ... whether it's a wolf! ... hunters! - And as if not honoring the embarrassed, frightened count with further conversation, he, with all the anger prepared for the count, hit the brown gelding on the sunken wet sides and rushed after the hounds. The count, as if punished, stood looking around and trying with a smile to arouse in Semyon regret for his position. But Semyon was no longer there: he, in a detour through the bushes, jumped a wolf from the notch. Greyhounds also jumped over the beast from two sides. But the wolf went into the bushes and not a single hunter intercepted him.Nikolai Rostov, meanwhile, stood in his place, waiting for the beast. By the approach and distance of the rut, by the sounds of the voices of the dogs known to him, by the approach, distance and elevation of the voices of those who arrived, he felt what was happening in the island. He knew that there were surviving (young) and seasoned (old) wolves on the island; he knew that the hounds had split into two packs, that they were poisoning somewhere, and that something bad had happened. He was always waiting for the beast on his side. He made thousands of different assumptions about how and from which side the beast would run and how he would poison him. Hope was replaced by despair. Several times he turned to God with a prayer that the wolf would come out on him; he prayed with that passionate and conscientious feeling with which people pray in moments of great excitement, depending on an insignificant cause. “Well, what does it cost you,” he said to God, “to do this for me! I know that You are great, and that it is a sin to ask You about it; but for the sake of God, make a seasoned one crawl out on me, and so that Karai, in front of the eyes of the “uncle”, who is looking out from there, slaps him with a death grip in the throat. A thousand times in that half-hour, with a stubborn, tense and restless look, Rostov cast a glance at the edge of the forests with two rare oaks over an aspen seat, and a ravine with a washed-out edge, and an uncle's hat, barely visible from behind a bush to the right.
“No, there won’t be this happiness,” thought Rostov, but what would it cost! Will not! I always, and in the cards, and in the war, in all misfortune. Austerlitz and Dolokhov brightly, but quickly changing, flickered in his imagination. “Only once in my life to hunt a hardened wolf, I don’t want more!” he thought, straining his hearing and eyesight, looking to the left and again to the right, and listening to the slightest nuances of the sounds of the rut. He looked again to the right and saw that something was running towards him across the deserted field. "No, it can't be!" thought Rostov, sighing heavily, as a man sighs when doing what he has long expected. The greatest happiness happened - and so simply, without noise, without brilliance, without commemoration. Rostov did not believe his eyes, and this doubt lasted more than a second. The wolf ran ahead and jumped heavily over the pothole that was in his path. It was an old beast, with a gray back and a reddish belly that was eaten. He ran slowly, apparently convinced that no one was watching him. Rostov looked round at the dogs without breathing. They lay, stood, not seeing the wolf and not understanding anything. Old Karay, turning his head and baring his teeth yellow teeth, angrily looking for a flea, snapped them on his hind thighs.
Chebyshev mechanism
Chebyshev mechanism is a mechanism that converts rotational motion into a motion close to rectilinear motion.
It was invented in the 19th century by the mathematician Pafnuty Chebyshev, who conducted research on the theoretical problems of kinematic mechanisms. One of these problems was the problem of converting rotational motion into motion approximated to rectilinear motion.
Rectilinear movement is determined by the movement of the point P - the midpoint of the link L 3, located in the middle between the two extreme points of the coupling of this four-link mechanism. ( L 1 , L 2 , L 3 , and L 4 are shown in the illustration). When moving along the section shown in the illustration, the point P deviates from the ideal rectilinear movement. The ratios between the lengths of the links are as follows:
Point P lies on the middle of the link L 3 . The given ratios show that the link L 3 is positioned vertically when it is in the extreme positions of its movement.
The lengths are related mathematically as follows:
Based on the described mechanism, Chebyshev made the world's first walking mechanism, which was a great success at the World Exhibition in Paris in 1878.
Other ways to convert rotational motion into approximately rectilinear motion are as follows:
- the Heuken mechanism is a variation of the Chebyshev mechanism;
- Lipkin mechanism - Posselier;
Notes
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| Mechanisms | |
|---|---|
| Rotational | |
| Rectilinear | |
| ...approximately | |
| Translational | |
| Compound movement | |
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- (named after the Russian mathematician and mechanic P. L. Chebyshev; 1821 1894) a flat 4-link hinged mechanism for reproducing the movement of a few points of the link (point M in the figure) in a straight line without the use of guides. Proposed in 1868. Used in ... ... Big encyclopedic polytechnic dictionary
This world's first walking mechanism, invented by a Russian mathematician, received general approval at the World Exhibition in Paris in 1878.
Pafnuty Lvovich Chebyshev is an outstanding Russian mathematician whose research a wide range scientific problems.
In his writings, he sought to combine mathematics with the foundations of natural science and technology. A number of Chebyshev's discoveries are related to applied research, primarily related to the theory of mechanisms. In addition, Chebyshev is one of the founders of the theory of the best approximation of functions using polynomials. He proved in general form law big numbers in probability theory, and in number theory - the asymptotic law of distribution of prime numbers, etc. Chebyshev's research was the basis for the development of new sections of mathematical science.
The future mathematician who became famous all over the world was born on May 26, 1821 in the village of Okatovo, Kaluga province. His father, Lev Pavlovich, was a wealthy landowner. The mother, Agrafena Ivanovna, was engaged in the upbringing and education of the child. When Pafnuty was 11 years old, the family moved to Moscow to continue teaching children. Here Chebyshev met some of the best teachers - P. N. Pogorevsky, N. D. Brashman.
In 1837 Pafnuty entered Moscow University. In 1841, Chebyshev wrote the work "Calculation of the roots of equations", and she was awarded that silver medal. In the same year, Chebyshev graduated from the university.
In 1846, Pafnuty Lvovich defended his master's thesis, and a year later he moved to St. Petersburg. Here he began teaching at St. Petersburg University.
In 1849, Chebyshev defended his doctoral thesis "Theory of Comparisons" (she was awarded the Demidov Prize). From 1850 to 1882 Chebyshev was a professor at St. Petersburg University.
A significant number of Chebyshev's works are connected with problems of mathematical analysis. Thus, the thesis of a scientist for the right to lecture is devoted to the integrability of some irrational expressions in algebraic functions and logarithms. Proof of the famous theorem on conditions for the integrability of a differential binomial in elementary functions set out in the work of 1853 "On the integration of differential binomials." Several other works by Chebyshev are devoted to the integration of algebraic functions.
In 1852, during a trip to Europe, Chebyshev got acquainted with the device of the steam engine regulator - J. Watt's parallelogram. The Russian scientist set out to "deduce the rules for the construction of parallelograms directly from the properties of this mechanism." The results of research concerning this problem were presented in the work "The Theory of Mechanisms Known as Parallelograms" (1854). This work simultaneously laid the foundations for one of the branches of the constructive theory of functions—the theory of the best approximation of functions.
In The Theory of Mechanisms, Chebyshev introduced orthogonal polynomials, which were later named after him. It should be noted that, in addition to approximation by algebraic polynomials, the scientist studied approximation by trigonometric polynomials and rational functions.
Later Chebyshev developed general theory orthogonal polynomials based on least-squares integration using parabolas, one of the methods of error theory used to estimate unknown quantities from measurements that contain random errors. This method is used when processing observations.
As a member of the artillery branch of the military scientific committee, Chebyshev solved a number of problems related to quadrature formulas - the results are presented in the work "On quadratures" (1873) - and the theory of interpolation. Quadrature formulas are used for the approximate calculation of integrals over the values of the integrand at a finite number of points.
Interpolation in mathematics and statistics is a method of finding intermediate values of a quantity from some of its known values.
Chebyshev's cooperation with the artillery department was aimed at improving the range and accuracy of artillery fire. Chebyshev's formula is known for calculating the range of a projectile. The works of Chebyshev had a significant impact on the development of Russian artillery science.
Chebyshev's research interest was attracted not only by Watt's parallelograms, but also by other hinged mechanisms. A number of the scientist’s works are devoted to their study: “On some modification of Watt’s cranked parallelogram” (1861), “On parallelograms” (1869), “On parallelograms consisting of any three elements” (1879), etc.
Chebyshev not only studied already existing mechanisms, but also designed them himself, in particular, he created the so-called "plantigrade machine", which reproduces the movements of an animal when walking, an automatic adding machine, mechanisms with stops, etc.
In 1868, Chebyshev proposed a special device - a flat four-link hinged mechanism for reproducing the movement of a certain point of a link in a straight line without the use of guides. This device was named after the Russian mathematician Chebyshev's parallelogram.
The scientist was also interested in the issues of cartography, the search for ways to obtain the optimal cartographic projection of the country, which allows the most accurate reproduction of the ratio of objects. Chebyshev's work "On the construction of geographical maps» (1856).
Chebyshev achieved significant success in solving the problem of distribution of prime numbers. He presented the results of his research in the works: "On the determination of the number of prime numbers not exceeding a given value" (1849) and "On prime numbers" (1852).
Pafnuty Lvovich Chebyshev was very interested in teaching. He organized a school of Russian mathematicians, whose graduates became famous mathematicians - D. A. Zolotarev, A. N. Lyapunov, K. A. Sokhotsky and others.
Further, in the work “On an Arithmetic Question” (1866), the scientist analyzed the problem of approximating numbers by rational numbers, which played a significant role in the development of the theory of Diophantine approximations. It should be noted that in number theory Chebyshev was the founder of a whole school of Russian scientists.
Chebyshev's works in this direction marked milestone in the development of probability theory. The Russian mathematician began to systematically use random variables, proved the inequality later named after him, developed a new technique for proving the limit theorems of probability theory, the so-called method of moments, and also substantiated the law of large numbers in a general form.
Chebyshev owns a number of works on the theory of probability. Among them are "An attempt at an elementary analysis of the theory of probability" (1845), "An elementary proof of one general position theory of probability” (1846), “On mean values” (1867), “On two theorems concerning probabilities” (1887). However, he failed to complete the study of the conditions for the convergence of the distribution functions of sums of independent random variables to the normal law. This was done by A. A. Markov, one of the scientist's students. Chebyshev's research in the field of probability theory was a significant stage in its development and became the basis for the formation of the Russian school of probability theory, which initially consisted of Chebyshev's students.
Chebyshev also worked on the theory of approximation. This is the name of the branch of mathematics that studies the possibility of an approximate representation of some mathematical objects others, usually of a simpler nature, as well as the problem of estimating the error introduced in this case.
Approximate formulas for calculating functions such as the root or constants were developed in antiquity.
However, the beginning modern theory approximation is Chebyshev's work "Sur les questions de minima qui se rattachent a la representation approximative des fonctions" (1857), which is devoted to polynomials that deviate least from zero, currently called "Chebyshev polynomials of the first kind."
Approximation theory has found application in the construction of numerical algorithms, as well as in the compression of information. There are currently several scientific journals, going to English language and devoted to problems of approximation theory: Journal on Approximation Theory (USA), East Journal on Approximation (Russia and Bulgaria), Constructive Approximation (USA).
Chebyshev made a great contribution to the development of artillery. Until now, textbooks on ballistics contain a formula derived by Chebyshev for calculating the range of a projectile.
For his merits, Chebyshev was elected a member of the St. Petersburg, Berlin and Bologna, Paris Academies of Sciences, a corresponding member of the Royal Society of London, the Swedish Academy of Sciences, etc. In addition, an outstanding mathematician was an honorary member of all universities in the country.
In the autumn of 1894, Chebyshev fell ill with influenza and soon died. However, the name of the outstanding Russian mathematician has not been forgotten to this day.
In 1944, the Academy of Sciences established the P. L. Chebyshev Prize.
Since the invention of the steam engine by James Watt, the problem has been to build a hinged mechanism that translates circular motion into rectilinear motion.
The great Russian mathematician Pafnuty Lvovich Chebyshev could not exactly solve the original problem, however, while investigating it, he developed the theory of approximation of functions and the theory of synthesis of mechanisms. Using the latter, he chose the dimensions of the lambda mechanism so that ... But more on that below.
Two fixed red hinges, three links are the same length. Due to its appearance, similar to Greek letter lambda, this mechanism got its name. The loose gray hinge of the small drive link rotates in a circle, while the driven blue hinge describes a trajectory similar to the profile of the hat white fungus.
On a circle along which the leading hinge rotates uniformly, we place marks at regular intervals and the marks corresponding to them on the trajectory of the free hinge.
The lower edge of the "cap" corresponds to exactly half the time of movement of the leading link around the circumference. Wherein Bottom part the blue trajectory differs very little from the movement strictly in a straight line (the deviation from a straight line in this section is a fraction of a percent of the length of the short leading link).
What else, besides a mushroom cap, does the blue trajectory look like? Pafnuty Lvovich saw a resemblance to the trajectory of a horse's hoof!
Let's attach a “leg” with a foot to the lambda mechanism. Attach to the same fixed axes in the opposite phase one more of the same. For stability, let's add a mirror copy of the already built two-legged part of the mechanism. Additional links coordinate their phases of rotation, and the axes of the mechanism are connected by a common platform. We have received, as they say in mechanics, a kinematic diagram of the world's first walking mechanism.
Pafnuty Lvovich Chebyshev, being a professor at St. Petersburg University, most spent his salary on the manufacture of invented mechanisms. He embodied the described mechanism "in wood and iron" and called it "The Walking Machine". This first walking mechanism in the world, invented by a Russian mathematician, received universal approval at the World Exhibition in Paris in 1878.
Thanks to the Moscow Polytechnic Museum, which preserved the Chebyshev original and made it possible for Mathematical Etudes to measure it, we have the opportunity to see an accurate 3D model of Pafnuty Lvovich Chebyshev's standing machine in motion.
Original articles by P. L. Chebyshev:
- On the transformation of rotational motion into motion along certain lines with the help of articulated systems / According to the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the Academy of Sciences of the USSR. 1948, pp. 161–166.
Museums and archives:
- The mechanism is stored in the Polytechnic Museum (Moscow); Department of Automation; PM No. 19472.
- Two wooden draft models of a plantigrade machine marked by P. L. Chebyshev are stored at the Department of Theoretical and Applied Mechanics of St. Petersburg State University.
Research:
- I. I. Artobolevsky, N. I. Levitsky. Mechanisms of P. L. Chebyshev / In the book: Scientific heritage of P. L. Chebyshev. Issue. II. Theory of mechanisms. - M.-L.: Publishing House of the Academy of Sciences of the USSR. 1945, pp. 52–54.
- I. I. Artobolevsky, N. I. Levitsky. Models of mechanisms of P. L. Chebyshev / In the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the Academy of Sciences of the USSR. 1948, pp. 227–228.
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In 1936, the Soviet engineer and scientist Vladimir Lukyanov created a computer in which all mathematical operations were performed by flowing water. Lukyanov's hydraulic integrator is the world's first computer for solving differential equations in partial derivatives - for half a century it was the only means of computing related to a wide range of problems in mathematical physics.
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