Paradoxical mechanism P. L

Since the invention of the steam engine by James Watt, the task has been to build a hinged mechanism that converts circular motion into linear motion.

The great Russian mathematician Pafnutiy Lvovich Chebyshev was unable to accurately solve the original problem, however, while studying it, he developed the theory of approximation of functions and the theory of synthesis of mechanisms. Using the latter, he selected the dimensions of the lambda mechanism so that... But more on that below.

Two fixed red hinges, three links have the same length. Because of its appearance, similar to the Greek letter lambda, this mechanism got its name. The loose gray hinge of the small driving link rotates in a circle, while the driven blue hinge describes a trajectory similar to the profile of a porcini mushroom cap.

Let us place marks at equal intervals on the circle along which the driving joint rotates uniformly and the corresponding marks on the trajectory of the free joint.

The lower edge of the “cap” corresponds to exactly half the time the driving link moves around the circle. In this case, the lower part of the blue trajectory differs very little from movement strictly in a straight line (the deviation from the straight line in this section is a fraction of a percent of the length of the short driving link).

What else, besides a mushroom cap, does the blue trajectory look like? Pafnuty Lvovich saw the similarity with the trajectory of a horse’s hoof!

Let's attach a “leg” with a foot to the lambda mechanism. Let's attach another one to the same fixed axes in the opposite phase. For stability, we will add a mirror copy of the already constructed bipedal part of the mechanism. Additional links coordinate their rotation phases, and the axes of the mechanism are connected by a common platform. We have received, as they say in mechanics, the kinematic diagram of the world's first walking mechanism.

Pafnutiy Lvovich Chebyshev, being a professor at St. Petersburg University, spent most of his salary on the manufacture of invented mechanisms. He embodied the described mechanism “in wood and iron” and called it the “Poligrade Machine.” This world's first walking mechanism, invented by a Russian mathematician, received universal approval at the World Exhibition in Paris in 1878.

Thanks to the Polytechnic Museum of Moscow, which preserved Chebyshev’s original and provided the opportunity for “Mathematical Etudes” to measure it, we have the opportunity to see in motion an accurate 3D model of Pafnutiy Lvovich Chebyshev’s plantigrade machine.

Original articles by P. L. Chebyshev:

• On the transformation of rotational motion into motion along certain lines using articulated systems / According to the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1948. pp. 161–166.

Museums and archives:

• The mechanism is kept in the Polytechnic Museum (Moscow); Automation Department; PM No. 19472.
• Two wooden draft models of a plantigrade machine with notes by P. L. Chebyshev are kept 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. Vol. II. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1945. pp. 52–54.
• I. I. Artobolevsky, N. I. Levitsky. Models of mechanisms by P. L. Chebyshev / In the book: Complete works of P. L. Chebyshev. Volume IV. Theory of mechanisms. - M.-L.: Publishing House of the USSR Academy of Sciences. 1948. pp. 227–228.

This world's first walking mechanism, invented by a Russian mathematician, received universal approval at the World Exhibition in Paris in 1878.

Pafnuty Lvovich Chebyshev is an outstanding Russian mathematician whose research covered a wide range of scientific problems.

In his works, he sought to combine mathematics with the fundamentals of natural science and technology. A number of Chebyshev's discoveries are associated with applied research, primarily related to the theory of mechanisms. In addition, Chebyshev is one of the founders of the theory of best approximation of functions using polynomials. He proved in general form the law of large 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 branches of mathematical science.

The future world-famous mathematician 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 involved in the upbringing and education of the child. When Paphnutius turned 11 years old, the family moved to Moscow to continue their children’s education. Here Chebyshev met some of the best teachers - P. N. Pogorevsky, N. D. Brashman.

In 1837, Paphnutius entered Moscow University. In 1841, Chebyshev wrote the work “Calculating the Roots of Equations,” and it was awarded a 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 dissertation “The Theory of Comparisons” (it 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 related to problems of mathematical analysis. Thus, the scientist’s dissertation for the right to give lectures is devoted to the integrability of some irrational expressions in algebraic functions and logarithms. The proof of the famous theorem on the conditions for the integrability of a differential binomial in elementary functions is presented in the 1853 work “On the integration of differential binomials.” Several more of Chebyshev's works are devoted to the integration of algebraic functions.

In 1852, during a trip to Europe, Chebyshev became acquainted with the device of the steam engine regulator - the J. Watt parallelogram. The Russian scientist set out to “derive the rules for the arrangement 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 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.

Subsequently, Chebyshev began developing a general theory of orthogonal polynomials based on integration using parabolas using the method of least squares - one of the methods of error theory used to estimate unknown quantities from measurement results that contain random errors. This method is used when processing observations.

As a member of the artillery department 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 to approximately calculate 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 based on 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, designed to calculate the flight range of a projectile. Chebyshev's works had a significant influence 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 a certain modification of Watt’s cranked parallelogram” (1861), “On parallelograms” (1869), “On parallelograms consisting of any three elements” (1879), etc.

Chebyshev not only studied 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-bar hinge mechanism for reproducing the movement of a certain point of the 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 issues of cartography and the search for ways to obtain an optimal cartographic projection of the country, allowing the relationships of objects to be reproduced as accurately as possible. Chebyshev’s work “On the Construction of Geographical Maps” (1856) is devoted to this problem.

Chebyshev made significant progress 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).

Pafnutiy Lvovich Chebyshev was very interested in teaching. He organized a school of Russian mathematicians, the graduates of which became famous mathematicians - D. A. Zolotarev, A. N. Lyapunov, K. A. Sokhotsky and others.

Further, in his 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 an entire school of Russian scientists.

Chebyshev's works in this direction marked an important stage in the development of probability theory. The Russian mathematician began to systematically use random variables, proved the inequality that was later named after him, developed a new technique for proving limit theorems in 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 probability theory. Among them are “An Experience in Elementary Analysis of the Theory of Probability” (1845), “Elementary Proof of a General Statement of the Theory of Probability” (1846), “On Average Values” (1867), “On Two Theorems Regarding Probabilities” (1887). However, he failed to complete the study of the conditions for the convergence of 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 possibilities of approximate representation of some mathematical objects by others, usually of a simpler nature, as well as the problem of estimating the error introduced by this.

Approximate formulas for calculating functions such as roots or constants were developed in ancient times.

However, the beginning of modern approximation theory is considered to be Chebyshev’s work “Sur les questions de minima qui se rattachent a la representation approximative des fonctions” (1857), which is devoted to polynomials that least deviate 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 information compression. Currently, several scientific journals are published in English and devoted to the 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. To this day, textbooks on ballistics contain the formula derived by Chebyshev to calculate the flight range of a projectile.

For his services, 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, the outstanding mathematician was an honorary member of all universities in the country.

In the fall of 1894, Chebyshev fell ill with the flu and soon died. However, the name of the outstanding Russian mathematician has not yet been forgotten.

In 1944, the Academy of Sciences established the P. L. Chebyshev Prize.

Municipal educational institution "Chudinovskaya secondary school" of Vyaznikovsky district

“Bringing theory closer to practice produces the most beneficial results, and it is not only practice that benefits from this; the sciences themselves are developing under its influence...”

P.L. Chebyshev

Questions of practice in creativity

P.L. Chebysheva

completed by: 9th grade student

Bedin Konstantin

Teacher: Dubrovina I.V.

Introduction……………………………………………………………………………….
1. The story of Chebyshev’s life and family
1.1. Chebyshev family…………………………………………………….
1.2. Childhood years of P.L. Chebysheva. The first teachers………………….
2. Scientific creativity of P.L. Chebysheva…………………………………….
2.1. Parallelogram of Chebyshev………………………………………..
2.2. Theory of best approximation of functions………………………
2.3. Formula P.L. Chebyshev for flat mechanisms……………..
2.4. Mechanisms P.L. Chebysheva………………………………………..
Bicycle mechanism
Press mechanism
Scooter chair mechanism
The "sorting" mechanism
Rowing mechanism
Mechanism of scales
Conclusion…………………………………………………………………..
Bibliography………………………………………………………….

Introduction

The scientific activity of P. L. Chebyshev was extremely diverse and fruitful. His main works relate to number theory, probability theory and mathematical analysis. In these areas, he discovered new research methods and left a number of important results. The originality of Chebyshev as a scientist is determined by the fact that he was able to connect the problems of mathematics with issues of natural science and technology and skillfully combine “abstract” theories with broad practice.

Thanks to his outstanding research in the field of mathematics, P. L. Chebyshev was elected a member of 25 different academies and scientific societies: St. Petersburg, Paris, Rome, Stockholm, Berlin, Bologna, Swedish Academies, the Royal Society of London, etc. President of the Paris Academy of Sciences, famous mathematician Charles Hermite stated that Chebyshev "is the pride of Russian science and one of the greatest mathematicians of Europe", and the Stockholm University professor Mittag-Leffler argued that Chebyshev is a brilliant mathematician and one of the greatest analysts of all time.

1. Life storyand the Chebyshev family

1.1. Chebyshev family

The information that has reached us about the Chebyshev family is very scarce. This family, being ancient, was not one of the famous ones. In the “Genealogy Book” N.I. Novikov there is an indication that the Chebyshevs received their surname from an ancestor nicknamed Chabysh, and it indicates that the Chebyshevs’ ancestors belonged to one of the tribes that in the distant past inhabited the eastern and southeastern parts of Russia.

In the 60s of the XVIII century. Pyotr Petrovich and Pavel Petrovich Chebyshev simultaneously served as officers in different guards regiments. Later, Pyotr Petrovich served as chief prosecutor of the Synod, but in 1774 he was dismissed, and his further fate is unknown. Pavel Petrovich - grandfather of P.L. Chebyshev - studied in the early 50s of the 18th century. at the Academic Gymnasium simultaneously with Ya.P. Kozelsky, who later became a famous mathematician and educator. After graduating from the gymnasium (1754), Pavel Chebyshev was assigned to military service - as an ensign in one of the guards regiments. In 1764, while still a young man, he retired and settled on his estate. Pavel Petrovich was distinguished by good health and, according to family legend, had a penchant for mathematics. He died at the age of 96, having stopped riding only two years before his death.

Fig.1. Lev Pavlovich Chebyshev

Pavel Petrovich had three children: daughter Pelageya and sons Lev and Peter. Pafnuty Lvovich's father, Lev Pavlovich Chebyshev (1789-1861), first served as a registrar in the Tula provincial government, then in 1812, with the rank of cornet of the Tula 1st Horse Cossack Regiment, he participated in the battle of Maloyaroslavets, Vyazma and Krasny, and in 1813. For courage in battles he received a military order. This is how L.P.’s feat is described. Chebyshev and several other officers of the Tula 1st Cavalry Cossack Regiment in the battles near Bautzen: “On the 9th, covering the battery and infantry retreat, and on the 10th being in a chain of riflemen, covering them, and then the entire retreat of the last chain until the night under strong grapeshot and rifle fire They encouraged their subcommanders with fire and excellent courage, in which they were completely successful.” (The battle took place on May 9-10, 1813). L.P. Chebyshev also took part in the capture of Paris by Russian troops in 1814.

In 1815, he retired and, like his father, devoted himself entirely to running the household. No information has been preserved about how he managed the farm and how he treated his serfs. But judging by the role that Lev Pavlovich played in Borovsky district, one must assume that he was a “gentleman of great hands”, far from any liberal and, even more so, revolutionary ideas.

Lev Pavlovich Chebyshev was popular among the nobles of the Borovsky district and was twice elected district leader of the nobility (from January 17, 1842 to December 5, 1847 and from December 12, 1856 to January 16, 1860). The reason for such popularity lay, apparently, not only in his organizational abilities and representative appearance; his secularism and hospitality played a significant role. Lev Pavlovich Chebyshev often organized balls in the noble assembly. To this day, in the State Historical Archive in Moscow, a painting has been preserved, painted by a Russian artillery officer in the 40s of the last century and depicting a ball in the city of Borovsk, organized in honor of Senator Davydov, who came to audit local institutions. In the foreground of this picture are the nobles of the county and the capital officials who accompanied Davydov, in the second are the local merchants. Among the nobles, in the very center of the picture, the mighty figure of Lev Pavlovich Chebyshev is visible.

The Chebyshevs had a mansion in Moscow, their own ride, and the horses they kept were so restive that only the coachman Savushka could handle them. They said about the latter that he was the illegitimate son of Lev Pavlovich and supposedly not the only one. Nevertheless, Lev Pavlovich was apparently a caring husband, evidence of this is found in the unpublished notes of the memoirs of Professor V.D. Shervinsky, who describes one of the characteristic episodes of the Chebyshev family moving from Moscow to Okatovo along a country road. This road was especially bad in bad weather due to numerous slopes. And Lev Pavlovich, running out of the heavy carriage and supporting it together with his servants, shouted: “Take care of the lady, take care of the lady above all else.” According to the recollections of those around him, Pafnuty Lvovich’s father was a good man. He was especially respected and loved by Pelageya Pavlovna’s daughter, Anna Ivanovna Shervinskaya.

The attitude towards Agrafena Ivanovna, the mother of Pafnuty Lvovich, was different. She belonged to the old noble family of the Poznyakovs, one of whose ancestors was “a Moscow nobleman and a centurion of the Moscow archers.” This family was numerous: its members are included in the genealogical books of the Smolensk, Kaluga, Nizhny Novgorod and Tver provinces. Agrafena Ivanovna had her own house in Moscow, near Prechistenka, on the corner of Long Lane. Lev Pavlovich and Agrafena Ivanovna Chebyshev lived in it continuously from 1832 to 1841, i.e. during the period of preparation of the two eldest sons (Pafnuty and Pavel) for entering the university and their stay there as students of the mathematical and law faculties.

According to documents and family legends, Agrafena Ivanovna appears to be “a stern woman, unloved by the people” for mistreating him. In his notes, Professor V.D. Shervinsky, for example, recalls: “Arriving in Moscow, my father left me with the Chebyshevs, who had their own house in Zubov. But I stayed not with the Chebyshevs themselves, but with Felitsata: she was either a housekeeper or just some kind of confidant of Agrafena Ivanovna, something like those persons whom the servants called “the lordly lady.” Agrafena Ivanovna did not accept me, because I was illegitimate and, therefore, according to the concepts of that time, I was certainly no match for such gentlemen as the Chebyshevs. But this Felitsata was probably allowed to receive and keep me.”

Fig.2. Agrafena Ivanovna Chebysheva

“When Agrafena Ivanovna came to visit us (outside the Butyrskaya outpost), I hid under the sofa so that she would somehow not see me, and it took a lot of effort for my mother to get me out of there; Yes, I don’t know if she succeeded. One thing is certain: I felt in my little heart the contemptuous attitude of this important landowner towards * and was in awe of meeting her. Fortunately for me, the Chebyshevs rarely visited us: family ties were recognized, but the difference in property was also not forgotten.”

In the Shervinsky family, the common noun “Poznyakovshchina” was used, with which they tried to convey a contemptuously lordly and arrogant attitude towards people who earned their living by their labor.

No information has been preserved about the attitude of Chebyshev’s parents towards their children. It is only known that they were people brought up in the spirit of their time, and personally supervised the initial education of their children. The latter were usually taught literacy by Agrafena Ivanovna, foreign languages ​​and arithmetic by Avdotya Quintillianovna Sukhareva, an educated girl who was a cousin of the young Chebyshevs and played the role of governess in their house. Lev Pavlovich and Agrafena Ivanovna gave their daughters an education that, according to the standards of that time, was considered decent for a noblewoman: to speak French as well as possible, to dance well, to know needlework and to be able to play the piano.

The eldest in age among the children of Lev Pavlovich and Agrafena Ivanovna Chebyshev was daughter Elizaveta, born on October 29, 1819. In 1852, she married P. L. Chebyshev’s former teacher Alexei Terentyevich Tarasenkov, which was, according to noble standards, a clear misalliance. This opinion did not change even after A. T. Tarasenkov became the director of the Sheremetevskaya (now named after N. V. Sklifasovsky) hospital and glorified his name, on the one hand, as a doctor who treated Gogol in the last days of his life and then described these days, on the other hand, as a physician-writer and prominent public figure. He died in 1873, leaving behind six children: three sons and three daughters. The eldest of Tarasenkov's sons, Alexey Alekseevich, was the caretaker of the Mariinsky Institute on Sofiyskaya Embankment in Moscow, where he occupied an apartment with his mother. Pafnuty Lvovich visited this apartment when visiting his older sister during his visits to Moscow. In the 80s, Elizaveta Lvovna already had granddaughters, who were especially interested in her famous brother. When meeting with them, Pafnuty Lvovich asked them about their studies, usually asked several questions about arithmetic, laughed when the children answered and added: “But I don’t know how to solve arithmetic problems.”

Pafnutiy Lvovich treated his older sister and her family very warmly. It was in this family that the most memories were kept about him: that he was very rich, but lived modestly and alone, had neither his own home nor his own travel, he usually drove a cab, never missing an opportunity to “bargain” with him, and even filled his own leaky galoshes.

All these family stories allowed I. A. Tarasenkov, the son of Elizaveta Lvovna, in 1922 to speak to members of the Society of Lovers of Old Moscow with memories of his famous uncle. The abstract of this report, which is the personal property of one of the surviving grandchildren of Pafnuty Lvovich, has been preserved to this day. In it, by the way, we read: “Father - Lev Pavlovich Chebyshev - landowner of Borovsky district, Kaluga province, respected local figure, powerful figure; mother - Agrafena Ivanovna, born Poznyakova, stern, not loved by the people. Children: Pavel, Paphnuty, Peter, Nikolai, Vladimir, Elizaveta, Ekaterina, Olga, Nadezhda. Legend of a dream. Sadness of parents, Preparation for university (32-37 years) in Moscow. Special features: modesty of life, frugality, buying up land (manager), preference for empty ones, praise for poor cultivation. The original monument at the birthplace."

Pafnuty Lvovich was born in 1821, two years later than his sister Elizabeth, and was the eldest of the brothers. In the metric book of Spas-na-Prognanyi of the Borovsky district of the Kaluga province it is written: “On May 4, 1821, in the village of Okatovo, a son, Pafnuty, was born to the landowner cornet Lev Pavlovich Chebyshev. Baptized on May 16. The successors were: Lieutenant Colonel Fedor Ivanov, son of Mitrofanov, from the nobility; girl Ekaterina Alekseeva, daughter of Zykov; prayed and baptized by priest Peter with the clergy.” It is very likely that Pafnuty Chebyshev received the rare name because 20 km from the village of Okatovo there was the Borovsky Pafnutyev Monastery, a monastery revered at that time by local residents.

Brother Pavel followed Paphnutius. At the same time as his older brother, he was preparing to enter Moscow University and then studied there at the Faculty of Law. Subsequently (from 1850 to 1856), with the rank of titular councilor, Pavel Lvovich was a judge of the Borovsky district court. He died early; no other information about him has been preserved.

Pafnuty Lvovich's second brother, Peter, was a military man, but retired early and was engaged in farming on his estate Kulage, Oryol province. He had four children: Lev, Paphnutius, Vladimir and Anna. The sons studied in the cadet corps, then at a military school and became officers in the guard. Pyotr Lvovich Chebyshev died during a train accident near Orel.

Nikolai (1830-1875) and Vladimir (1832-1905) are the youngest brothers of Pafnuty Lvovich. Both graduated from the artillery school and the academy and were hired as mathematics tutors at the academy on the recommendation of M.V. Ostrogradsky. Subsequently, Nikolai Lvovich, with the rank of colonel, was the head of the Warsaw training ground and did a lot to improve this important area in artillery. He died with the rank of major general in 1875 as head of the Kronstadt fortress artillery.

Vladimir Lvovich, an artillery general, an outstanding artillery scientist, was an emeritus professor of the Artillery Academy, the founder and first editor of the “Armory Collection”, the founder of the cartridge and gun business in Russia, as well as the founder of the doctrine of surface properties. In 1874, he was the first to study the process of cylindrical cutters and establish the main reasons causing micro-roughness on the machined surface. The conclusions he made found practical application at the Tula plant and were used in theoretical works of this period. They have not lost their meaning to this day. Here is one of the assessments he gave today. “The founder of the scientific direction in the study of micro-irregularities of surfaces processed by cutting is the Russian scientist, Professor V.L. Chebyshev, who in 1873 completed a detailed theoretical study of the cylindrical milling process, the results of which he reported in November 1874 to the St. Petersburg branch of the Russian Technical Society.

Many of the provisions that the researcher came to in his work “On the most advantageous way to use roller cones and roller-cone machines” are of global significance. In putting forward these provisions, V.L. Chebyshev pointed out that the dimensional accuracy of the processed part depends on the height of the scallops formed on the surface. As a result of analyzing the milling conditions, V.L. Chebyshev derived an equation for determining the height of microroughnesses.”

It is worth dwelling on the research of P. L. Chebyshev on the theory of the locking mechanism. The lock of a gun has, in comparison with the barrel, if not a greater, then at least no less influence on the efficiency of shooting. Before V.L. Chebyshev, the lock of a gun was written about in every work on shooting, but in passing, in the most superficial way, V.L. Chebyshev drew attention to the importance of the locking mechanism of a gun and, having studied this issue, came to the results that formed the basis gun lock theory.

Of the brothers, Vladimir Lvovich was closest to Pafnutiy Lvovich. He also witnessed the last days of his life. With the financial support of V.L. Chebyshev in 1899-1907. The first two-volume collected works of P. L. Chebyshev was published. After the death of Pafnuty Lvovich, Vladimir Lvovich transferred his correspondence with Russian and foreign scientists, a portrait, mathematical manuscripts and models to the Academy of Sciences. To store the latter, he ordered a special cabinet, which is currently located at the Mathematical Institute. V. A. Steklov of the USSR Academy of Sciences.

Conveying the listed legacy, V.L. Chebyshev wrote to the permanent secretary of the Academy of Sciences N.F. Dubrovin: “I consider it necessary to communicate the will of the deceased, which he expressed many times and for the exact execution of which I consider it my duty to ask on behalf of myself and the nephews of the deceased. This will lies in the fact that such manuscripts of his can be printed in their entirety on which he made the inscription: “You can print.”

Ekaterina Lvovna, the younger sister of Pafnutiy Lvovich, married Mikhail Nikolaevich Lopatin, a famous lawyer who was extremely popular in Moscow society and served as chairman of the department of the Moscow Court Chamber. They had children: Nikolai Mikhailovich - a collector of Russian songs; Lev Mikhailovich - a famous philosopher of the idealistic trend, professor at Moscow University, author of "Positive Problems of Philosophy", an active contributor to the journal "Questions of Philosophy and Psychology"; Alexander Mikhailovich - prosecutor; Vladimir Mikhailovich - an artist of great talent who played on the stage of the Moscow Art Theater in the 20s x years of our century, Ekaterina Mikhailovna - writer.

The Lopatin family was one of the highly cultured Moscow families, where prominent Russian figures often visited: I. S. Aksakov, A. F. Pisemsky, S. M. Solovyov, I. E. Zabelin and others. Information about how often Paphnutius visited Lvovich during his visits to Moscow with Ekaterina Lvov and how he treated the Lopatins has not been preserved.

Olga Lvovna Chebysheva was married to one of the Goncharovs, from whose family Natalya Nikolaevna, Pushkin’s wife, came.

She lived at the Polotnyany Factory, the family name day of the Goncharovs, and was considered “the primary heir of A.S. Pushkin,” as stated in the summary of P.A. Tarasenkov’s report on old Moscow. By the way, she wrote the historical story “The Year One Thousand Eight Hundred and Twelve” (M., 1867), published by the “Society for the Distribution of Useful Books.

Chebyshev's youngest sister was Nadezhda, who married M.P. Zakharov and had children. She was very concerned about preserving family legends and traditions and was one of all the Chebyshevs who constantly visited the village of Okatovo. Nadezhda Lvovna, more than other sisters, kept in touch with Chebyshev, often traveled from her own estate Rudakovo (Vorovsky district) to St. Petersburg and visited her famous brother, whom she treated very respectfully. Chebyshev himself did not visit Rudakovo, but at the invitation of Nadezhda Lvovna, his daughter occasionally visited there along with her husband, Colonel Leer, and her own daughter. Chebyshev was not officially married, but had a daughter, whom, according to relatives, he provided for well, but did not adopt and, apparently, never lived with her. In the 80s of the last century, according to reviews of people who knew her, she was a petite, beautiful and elegant lady with signs of considerable spoilage. The Leer family usually stayed in Rudakovo for several days and returned to St. Petersburg.

The Chebyshev brothers were rich because they inherited large and profitable estates from their parents: Peter and Vladimir in the Oryol province, Paphnutius in Kaluga, etc.

Pafnutiy Lvovich had considerable income from his position as an academician and professor, as well as from the publication of his scientific works. Having relatively large amounts of money, Pafnuty Lvovich used part of it to purchase land. This operation was carried out by its manager, who profitably resold the purchased, mostly empty or poorly cultivated lands.

Chebyshev did not do this for reasons of his own profit. The fact was that his sisters received a significantly smaller inheritance than he himself received from his brothers. And being one of the eldest in the Chebyshev family, he considered it his duty to increase their share at the expense of the lands donated to him. So, in the Tula province, he bought the former estate of M. Yu. Lermontov Kropotovo and gave it to Elizaveta Lvovna, and Nadezhda Lvovna - the Lokotsi estate bought there; At the end of his life, he gave her the Okatovsky house that belonged to him.

According to information that has reached us, all members of the family of Lev Pavlovich and Agrafena Ivanovna Chebyshev were very conservative and monarchist-minded, especially Pyotr Lvovich Chebyshev.

Only the family of Alexei Terentyevich and Elizaveta Lvovna Tarasenkov was distinguished by democracy.

To characterize the environment of Pafnuty Lvovich, information about Dmitry Ivanovich and Aiva Ivanovna Shervinsky is not without interest. They were the children of Pelageya Pavlovna Chebysheva, who married the staff doctor Shervinsky. A few words about Pelageya Pavlovna herself, Pafnutnya Lvovich’s aunt. She had an independent, strong, what is called a masculine character. Because of her misalliance, Pelageya Pavlovna did not receive everything that was due to her by inheritance. But she managed to upset what she received so that her children, unlike the children of her brother, Lev Pavlovich, lived far from contentedly.

In different provinces, she managed to acquire pieces of estates by rumor, for the sole reason that, when marrying off her daughters, she would have the right to talk about a dowry of two or three estates for each of them.

In relation to Pelageya Pavlovna's numerous children, Pafnuty Lvovich was closest to Dmitry Ivanovich and Anna Ivanovna Shervinsky. The first served first in the Life Guards Cuirassier Regiment, but not for long, since maintenance in this brilliant regiment was beyond the means of his parents. Then he transferred to the army cavalry, but soon retired due to illness. After that, he served in Siberia, first as the “manager of the salt section”, and then as the “manager of the IV department of the Main Directorate of Western Siberia.” In the early 50s, Dmitry Ivanovich, leaving his son Vasya in Moscow, in the family of his uncle, Lev Pavlovich Chebyshev, moved to St. Petersburg, but fell ill there and died.

Pafnuty Lvovich visited his cousin in the hospital. He buried him, as he told Anna Ivanovna Shervinskaya in two letters that have survived to this day. Here are their contents.

“To my extreme regret, I must tell you, dear sister, unpleasant news. About two weeks ago, my brother, Dmitry Ivanovich, felt double objects in his eyes, went to consult with Arend and, on his advice, began to take medicine. After that, he felt heaviness in his stomach, weakness in his body, and invited a doctor to come to him. He lived not far from me - the famous hotel Heide, and we saw him almost every day.

On Thursday the 14th, his weakness became so severe that he considered it best to go to the Mary Magdalene hospital, which is near the Tuchny Bridge; This hospital is very close to the Heide Hotel; and the doctor of that hospital used it.

He went to the hospital so soon that I only found out when he was there. On Thursday, the 15th, I visited him, along with a doctor from the hospital who was familiar to me: my brother complained of weakness, pain in his side, heaviness in his head; the doctor told me that he had a blockage, but not dangerous; his head is not good: he seemed to be talking. I left him in this position on Thursday, and on Friday - at 5 o'clock in the morning - he was gone. Now is his funeral - he will be buried at the Smolensk cemetery. I will collect the things he left at the hospital and hotel and send them to you. And you take care of the fate of Vasya’s pupil, who now lives with a man in our house.

Your most humble servant Pafnutny Chebyshev.

According to brother Dmitry Ivanovich, I think that he should still have silver and a gun, quite chained, he spoke about them as his securse: take measures so that this is not stolen.

My address: In St. Petersburg on Vasilievsky Island in the eleventh line, between Bolshoi and Sredny Avenues: - Transchel's house.

Another letter:

For some time I could not muster the courage to begin sorting through the papers and things of my late brother: every thing reminded me so vividly of him. Finally, I made up my mind and found a paper about Vasya: this is the condition with his mother, according to which he was raised by his brother Dmitry Ivanovich; You will find this paper in a bag with other papers; On the back I wrote: here are Vasya’s documents. In addition to this sack, a box is sent to you. In it, in the pocket of that thing that you call indescribable, in handkerchiefs and paper there is a watch - take it out carefully. At the bottom of the box you will find a gift from me to you and to several of our closest relatives: for whom there are inscriptions. For delivery by signature, you can send them to Pyotr Timofeevich. In addition to the bag and the box, an overcoat, a gray coat and a pair of boots are sent to you - this is not included in the box. Then, due to inconvenience, it remained to send: 1) a pillow, 2) hats, 3) a pound of sugar and a pound of Kaletov candles, 4) a chibouk with a pipe. These things will remain until our date or until the opportunity becomes especially convenient. Now about the fur coat and money. You write that my money is 100 rubles, much less. Here's the score - friendship doesn't lose track.

At the hotel, according to the bill, which was made while my brother was still alive, - 19 p. 50 k. ser.

His lackey's calculation is 6 rubles.

Added to the remaining money of the deceased 23 for the funeral - 19 rubles.

Total 44 rub. 50 kopecks

Judging by the time, I hope to sell the fur coat at a profit, and then I’ll send you the rest of the money; and, perhaps, you will find a hunter for her in Moscow: in any case, it is impossible to send her with Fyodor: I am afraid that he will lose this too. You don’t have to give him anything for delivery - he will receive 3 rubles from me.

Your brother P. Chebyshev.

Regarding these letters and especially the accounts in them, Professor V.D. Shervinsky writes in his memoirs: “Pafnuty Lvovich, a famous mathematician in the future, a member of the Russian and French academies of sciences, lived then in St. Petersburg and probably visited my father in hospital; He buried him, sending Anna Ivanovna Shervinskaya a letter with the imposition of funeral expenses and an inventory of the remaining insignificant property. I will note here by the way that I was very happy. when, having become a doctor, I was able to pay Pafnuty Lvovich the money he spent on my father’s funeral.”

L.I. Shervinskaya, to whom the above letters from Chebyshev were addressed, was a public figure, one of the first in Russia at that time, a full member of the Moscow Society of Agriculture, and was awarded a medal for successful experiments in breeding silkworms in central Russia.

After the death of her parents, Anna Ivanovna lived for a long time in Okatovo, the estate of her uncle Lev Pavlovich Chebyshev. Her early years passed there, and there she received her meager education. Not wanting to lead the life of a hanger-on with wealthy relatives, Anna Ivanovna got a job as a caretaker of one of the orphanages in Moscow.

Having received notice from P.L. Chebyshev in October 1853 about the death of her brother, A.I. Shervinskaya decided to take in her three-year-old nephew Vasya, who was left alone in Moscow with the family of Lev Pavlovich and Agrafena Ivanovna Chebyshev. The latter often visited A.I. Shervinskaya later, and she really appreciated this. Pafnutny Lvovich also visited her while in Moscow. On one of these visits, Shervinskaya turned to him with the following question: “Please tell me, Pafnutny, what should I give Vasya to read? The boy is inquisitive, reads willingly and keeps asking what he should read.” Pafnuty Lvovich thought, somewhat puzzled by this question, and answered: “You know what, sister, let him read Karamzin’s “History of the Russian State.”

Note that in the first half of the 19th century. Karamzin’s “History of the Russian State” was considered an outstanding book, and many famous Russian people of that time had dear childhood memories associated with it. From this book they got acquainted with what happened in ancient years and learned to love their Motherland. The great talent and hard work with which the book was written made a deep impression on Chebyshev. That is why, in our opinion, he advised the boy Vasya Shervinsky to read “The History of the Russian State.” “This advice,” writes V.D. Shervinsky in his memoirs, “was not carried out, and it is unlikely that even if we had gotten Karamzin, I would have been able to complete such a serious essay at this age.”

One curious phrase is known from Vasily Dmitrievich Shervinsky, once said by Chebyshev: when asked whether he, as a member of the French Academy of Sciences, was going to visit Paris, he answered in the negative, adding: “there is no need to spoil them too much.”

1.2. Childhood years of P.L. Chebysheva. First teachers

Pafnuty Lvovich Chebyshev, unfortunately, did not leave behind any memories, much less autobiographical notes. Only in 1853 did he provide brief information about himself to Poggendorff for the Biographical and Literary Dictionary. They were used by A. M. Lyapunov when compiling an essay about P. L. Chebyshev. Less is known about the childhood and adolescence of the great Russian scientist. When at the beginning of the 20th century. This information was required by K. A. Posse, but among the surviving relatives of Pafnuty Lvovich there was not a single one who could give it. Vladimir Lvovich Chebyshev was much younger than his brother and could not tell anything about the first years of Pafnuty Lvovich’s life.

Lev Pavlovich and Agrafena Ivanovna Chebyshev with their large family lived almost constantly on the Okatovo estate, in a large wooden house of simple architecture, with a balcony and stairs to the garden. The house and garden were located along the descent to the Istya River, which flows into Paru. The house had large formal rooms, furnished with antique furniture and overlooking the garden. There was a billiard table in the hall, and in the living room there were two glass slides with souvenirs, among which a shako, testifying to military ancestors, attracted attention. In the sofa room there is an antique harpsichord, in the bedroom there is a huge four-poster bed. Next to the bedroom there was a prayer room with many ancient icons; Old Believers came there to pray (the surrounding villages were schismatic at that time).

Fig.3. The Chebyshev House on the Okatovo estate

To date, only the following is known about Chebyshev’s childhood. He learned to read and write from his mother, and French and arithmetic from his cousin, Avdotya Quintillianovna Sukhareva, a very educated girl who apparently played an important role in Chebyshev’s upbringing. Pafnuty Lvovich kept her portrait for the rest of his life.

Remembering his childhood, Chebyshev, according to D.I. Mendeleev, said that he owed his development to his former music teacher. who didn’t teach him music, but trained the child’s mind to precision and analysis.

At the age of 10, Chebyshev and his uncle, Pyotr Pavlovich, made their first long trip to the Caucasus, visiting Zheleznovodsk, Pyatigorsk and other places. Since childhood, he had one leg cramped, limped a little and walked with a stick. It is still not possible to find out the cause of this physical defect, which played a large role in the life of Pafnuty Lvovich. This shortcoming was a source of sadness for his parents, who wanted to see their eldest son become an officer. He caused a lot of grief to Pafnuty Lvovich himself, forcing him to avoid children's games and forcing him to stay at home more. True, the boy did not sit idly at home, but devoted himself with great love to the construction of mechanical devices. Finally, partly thanks to this shortcoming, Pafnutvy Lvovich became a student and not an officer. In the mentioned report by P. L. Tarasenkov, the “legend of sleep and the sadness of parents” is connected with this deficiency. What kind of legend this is - but we managed to establish it. As for the “parents’ sadness,” it is understandable without further explanation.

Chebyshev's first teacher in mathematics - gymnasium inspector P. I. Pogorelsky - was distinguished by his harsh treatment of students and his predilection for punitive measures. Always serious, with a frowning face, abrupt speech, demanding to the point of pedantry, not leaving a single student’s offense without a stern remark, reprimand or punishment. P. N. Pogorelsky kept his students (and not only his students) in the strictest subordination to himself.

Platon Nikolaevich Pogorelsky (1800-1852) in the early 30s was considered one of the best and most famous teachers in Moscow. At this time (1832), Lev Pavlovich Chebyshev brought his eldest sons, Pafnuty and Pavel, from the village of Okatovo to Moscow. Having decided to educate them at home, L.P. Chebyshev invited Pogorelsky, a master of Moscow University, to join them as a teacher of mathematics and physics.

Pogorelsky combined experience with activity, energy with perseverance, justice with exactingness, love for his pets with exactingness, sometimes bordering on cruelty. As a mathematics teacher, Pogorelsky was famous for his unusual ability to keep the entire class in constant tension during a lesson and to present his science in a clear and accessible form.

Having become the director of the gymnasium. Pogorelsky in a short time gave it an exemplary device. When public schools were transferred to his jurisdiction as the director of the gymnasium in 1841, he, through a series of successfully taken measures, was able to quickly raise primary education to such a height that attracted the attention of the Minister of Public Education and forced him to demand that all other schools be structured according to the model Moscow P. N. Pogorelsky selected teachers for his gymnasium very carefully.

Pogorelsky increased his fame as an outstanding teacher by publishing manuals on mathematics. Not finding contemporary educational and mathematical literature, both translated and original, a textbook that corresponded to his views and pedagogical requirements, he translated from French in the early 30s “A Course in Pure Mathematics...” (M., vol. 1, 1832 ; t. 2, 1833; t. 3. 1834). This translation was so successful and so well adapted to the gymnasium curriculum in mathematics that in a relatively short time it went through numerous editions and was adopted as a textbook for gymnasiums (especially “Algebra”, published in the 8th edition in 1863).

We see, therefore, that Pogorelsky sought to improve methods of teaching elementary mathematics and textbooks on this science. He first of all implemented all his achievements in this direction in the gymnasium entrusted to him. And it is no coincidence that the students of this gymnasium almost until the end of the 19th century. showed some special attraction to mathematics: their success in this subject was higher than in others, and most of those who completed the course chose the mathematics department for their further education.

What goes around comes around. And we believe that the first seeds of love for mathematics, for a concise, clear and accessible imposition of its foundations, rigor and high demands on one’s own knowledge and on the knowledge of others - all this was sown in Chebyshev’s mind by Pogorelsky.

Chebyshev studied elementary mathematics using his textbooks, since at that time they were the most popular and were republished almost 2-3 years later. These textbooks successfully combined completeness of content with clarity and conciseness of presentation. Chebyshev appreciated Pogorelsky's textbooks when, already a member of the Scientific Committee of the Ministry of Public Education on Mathematical Sciences, he recommended them, mainly Algebra, as educational manuals for gymnasiums. About this textbook by Pogorelsky, Chebyshev, by the way, said that this is the best of all books in Russian, because it is “the most concise.”

Pogorelsky's "Geometry" was less popular than his "Algebra", but in some educational districts (for example, Moscow) it was used as a guide for a long time.

Information has reached us about another teacher of P. L. Chebyshev - A. T. Tarasenkov.

A. T. Tarasenkov was the son of a small fur trader; he studied at the 1st Moscow gymnasium in the early 30s, which he was forced to leave due to domestic circumstances. His parents assigned him to work in one of the Moscow private stores on the Nozhevaya Line in the Trading Rows. Thanks to a happy accident, which he owed to the inspector of the 1st gymnasium P. N. Pogorelsky, Tarasenkov returned to the gymnasium again, successfully graduated from it and then entered the medical faculty of Moscow University. Among the students, he stood out for his excellent knowledge of the Latin language: he not only easily translated Latin classics, but also spoke this ancient language fluently, knew many Latin riddles and sayings, using them without any difficulty.

As an excellent Latinist, Tarasenkov was known to the Moscow public, including Chebyshev's parents, who invited him as a home teacher for their eldest sons. This is how Tarasenkov’s first acquaintance with Pafnuty Lvovich took place. It should be noted that the 30s of the last century were the years when classicism in the educational system reached its greatest power. Ancient languages ​​were given one of the first places in gymnasiums and universities. Therefore, the concern that Chebyshev’s parents showed when they were faced with the question of teaching their eldest sons the Latin language is understandable.

Pafnutiy Lvovich passed the university exam in this language, as well as in other subjects, very successfully. He owes this success largely to medical student Tarasenkov, one of his first mentors.

Pafnutiy Lvovich entered Moscow University at the age of 16. The young man immediately discovered a huge talent in mathematics. While still a student, he received a silver medal for his essay “Calculating the Roots of an Equation,” and in 1846 he defended his master’s thesis “An Experience in Elementary Analysis of Probability Theory.” In 1847, the young scientist was invited to work at St. Petersburg University, where he worked for 35 years. Here in 1849 he defended his doctoral dissertation “The Theory of Comparisons,” which was awarded the Demidov Prize by the St. Petersburg Academy of Sciences. In 1850, Chebyshev was elected professor. He was entrusted with giving lectures on analytical geometry, number theory, higher algebra, etc. Soon Chebyshev became an adjunct at St. Petersburg University. At the same time, he is engaged in scientific work at the Russian Academy of Sciences. Since 1856, Pafnuty Lvovich has been an extraordinary, and since 1859, an ordinary academician of the St. Petersburg Academy of Sciences.

For forty years, Chebyshev took an active part in the work of the military artillery department and worked to improve the range and accuracy of artillery fire. In ballistics courses, Chebyshev's formula for calculating the flight range of a projectile has been preserved to this day. With his works, Chebyshev had a great influence on the development of Russian artillery science.

2. Scientific creativity of P.L. Chebysheva

As the most important feature of the scientific creativity of P.L. Chebyshev should be noted for his constant interest in practical issues. This interest was so great that, perhaps, it largely determines the originality of P.L. Chebyshev as a scientist. It is no exaggeration to say that most of his best mathematical discoveries were inspired by applied work, in particular, his research on the theory of mechanisms. The presence of this influence was often emphasized by Chebyshev himself, both in mathematical and applied works, but he most fully expressed the idea of ​​​​the fruitfulness of the connection between theory and practice in the article “Drawing Geographical Maps”: “Bringing theory closer to practice gives the most beneficial results, and not Practice alone benefits from this; The sciences themselves develop under its influence: it opens up new subjects for them to study, or new aspects in subjects that have long been known. Despite the high degree of development to which the mathematical sciences have been brought by the works of the great geometers of the last three centuries, practice clearly reveals their incompleteness in many respects; it proposes questions that are essentially new for science and, thus, calls for the exploration of completely new methods. If a theory gains a lot from new applications of an old method or from new developments of it, then it gains even more from the discovery of new methods, and in this case science finds itself a faithful leader in practice.”

Among the huge number of tasks that his practical activity poses to a person, one is of particular importance: “How to manage your means to achieve the greatest possible benefit.” That is why “most of the questions of practice are reduced to problems of the largest and smallest magnitudes, completely new to science, and only by solving these problems can we satisfy the requirements of practice, which everywhere seeks the best, the most profitable.” The above quote is for P.L. Chebyshev was the program for all his scientific activities and was the guiding principle of his work.

Numerous applied works by P.L. Chebyshev, bearing far from mathematical titles - “On one mechanism”, “On gear wheels”, “On the centrifugal equalizer”, “On the construction of geographical maps”, “On cutting dresses” and many others - were united by one basic idea - how to place cash to achieve the greatest benefit.

Thus, in his work “On the Construction of Geographical Maps,” he sets out to determine a projection of the map of a given country for which the scale distortion would be minimal. In his hands this problem received a comprehensive solution. For European Russia, he brought this decision to numerical calculations and found that the most advantageous projection would give a scale distortion of no more than 2%, while the projections accepted at that time gave a distortion of at least 4-5%.

2.1. Chebyshev parallelogram

He spent a significant portion of his efforts on the design (synthesis) of hinged (joint, as Chebyshev said) mechanisms and on the creation of their theory. He paid special attention to improving the Watt parallelogram mechanism, which serves to transform circular motion into rectilinear motion. The point was that this mechanism, fundamental for steam engines and other machines, was very imperfect and gave, instead of rectilinear movement, curvilinear movement. This substitution of one movement for another caused harmful resistances that spoiled and wore out the machine. Seventy years have passed since Watt's discovery. Watt himself, his contemporaries and subsequent generations of engineers tried to combat this defect, but, groping, through trials, they could not achieve significant results. P.L. Chebyshev looked at the matter from a new point of view and posed the question this way: to create mechanisms in which the curvilinear movement would deviate as little as possible from the rectilinear one, and at the same time determine the most advantageous dimensions of the machine parts.

A hinge mechanism proposed by P. L. Chebyshev in 1868 to reproduce the movement of a certain point of the mechanism in a straight line. Chebyshev parallelogram is a flat articulated four-bar ABCD (rice. 4 ), also called a straight-line guide mechanism, in which the lengths of the links satisfy the relation 3 d–a= 2b. Length of an approximately straight section of a point's trajectory M gets bigger as you increase AB, but at the same time the deviation from straightness increases. Chebyshev parallelogram shown in Fig. solid lines, in the middle position it resembles the Greek letter λ and is therefore called λ-shaped. Chebyshev also indicated another modification of this mechanism AB 1 C 1 D 1 (Fig. 4). In this modification, called cross, the trajectory of the point M coincides with the trajectory of the same point in the λ-shaped mechanism, and the lengths of the links are related by the relations: AB 1 = C 1 D 1 = 2b, B 1 C 1 = 2a, B 1 M = a, AD 1 = 2d. The Chebyshev parallelogram is also known, in which the angle between the lines NE And CM differs from 180. The Chebyshev parallelogram is used in devices to obtain the rectilinear motion of a point without guides.

The famous “Chebyshev parallelogram” has received practical application in the navy in artillery fire control systems.

2.2. Theory of best approximation of functions

It is especially important for the history of mathematics that the design of mechanisms and the development of their theory served P.L. Chebyshev was the starting point for the creation of a new branch of mathematics - the theory of best approximation of functions by polynomials. Using the apparatus of the theory of functions that least deviate from zero, which he specially developed, he showed the possibility of solving the problem of approximately rectilinear motion with any degree of approximation to this motion. Here P.L. Chebyshev was a pioneer in the full sense of the word, having absolutely no predecessors. This is an area where he worked more than in any other, finding and solving more and more new problems and creating a new extensive branch of mathematical analysis with the totality of his research, which continues to develop successfully even after his death. The initial and simplest formulation of the problem began with the study of Watt's parallelogram and consisted in finding a polynomial of a given degree that would deviate from zero less than all other polynomials of the same degree in a certain given interval of change in the argument. Such polynomials P.L. Chebyshev were found; they were later called "Chebyshev Polynomials". They have many significant properties and currently serve as one of the most commonly used research tools in many questions of mathematics, physics and technology.

Chebyshev devoted a lot of work to improving interpolation, which is of great importance in astronomy, physics, chemistry and in general in all applied and experimental sciences.

It can be said without exaggeration that a significant part of the general conclusions in the experimental sciences represents, in essence, only the interpretation of various kinds of interpolation formulas.

People have become so accustomed to the idea of ​​interpolation that they often forget its purpose as a method of approximate calculation, and the conclusions obtained from interpolation formulas are sometimes passed off almost as laws of nature.

The most important is interpolation using polynomials, which has occupied scientists for a long time.

The problem of interpolation was considered by Wallis (300 years ago), then by Newton, who laid the foundation for the theory and gave a special formula that is still used today, by Stirling, Euler, Cauchy, Lagrange, Gauss, Bessel and many other first-class geometers.

In commonly used interpolation formulas, the degree n of the interpolating polynomial is set in advance and one below the number of given values ​​of the interpolated function.

When interpolating using the least squares method, which is inevitable when the number of given function values ​​is small, it is necessary to perform a very large number of multiplications and divisions, sometimes multi-digit numbers (addition and subtraction no longer count).

For example, with n = 3, generally speaking, you need to perform about 120 such operations, and, in addition, a lot of tedious calculations to determine the value of the quadratic error.

If the latter turns out to be unsatisfactory, it is necessary to obtain from observations a larger number of values ​​of the interpolated function and, when constructing a new interpolating polynomial of the highest degree and determining the corresponding error, perform a series of new, even more tedious calculations, the number of which quickly increases with increasing degree of the polynomial. So, with n = 5, it will be necessary to perform about 5000 multiplications and divisions, not counting those needed to determine the quadratic error. Moreover, by increasing the degree of the interpolating polynomial, we do not know in advance to what extent we are approaching the interpolated function and whether we are approaching it at all.

On the contrary, sometimes increasing the degree of a polynomial may not be able to increase the degree of approximation as desired, as shown, for example, by Prof. Runge using a simple example of a function
, interpolated by the Lagrange method

Chebyshev could not come to terms with such shortcomings in an issue so important for both theory and practice, and undertook a series of research in this area.

He approached the problem from a new, extraordinary point of view, guided, as always, by the same general idea: to extract the greatest possible benefit from the data of reality.

He posed the problem as follows: given n+1 values ​​of a function for given values ​​of an independent variable, find its value for some other value of the variable, let’s say x, under the guise of a polynomial of degree m not exceeding the number n, so that the errors of these function values ​​have the smallest influence on its calculated value at x.

Such an original formulation of the question required no less original ingenuity to create an appropriate method for solving it.

Chebyshev’s insightful mind found the source of this method in the theory of continued fractions in connection with the foundations of probability theory, and the memoir itself, where he developed his interpolation method, was entitled: “On continued fractions.” In general, I note by the way, Chebyshev widely used the theory of continued fractions and gave a number of remarkable applications, the scope of which was then expanded by his followers: Academician A. A. Markov, our honorary member K. A. Posse, etc.

In this way, he obtained a new, general interpolation formula, which significantly eliminated the shortcomings of previous methods and at the same time opened a wide field for new conclusions in many other areas of analysis.

In the Chebyshev formula, the number of terms of the interpolating polynomial is not specified in advance, but they are determined one after another sequentially, without resorting to the tedious solution of sets of many equations, as in many other methods.

The number of multiplication and division operations is greatly reduced.

For example, with a polynomial degree m = 3 and n = 4 (the most difficult case for Chebyshev), the number of these actions is only 41, while with other methods it can exceed 120. With n = 6 and m = 5 this number is no more than 107, and with conventional techniques it can reach, as mentioned, up to 5000.

As the number t increases, the difference becomes even more impressive.

Moreover, with the Chebyshev method, each time, when sequentially calculating the terms of a polynomial, the quadratic error is also calculated, which immediately indicates whether it is necessary to calculate the next term or whether it is enough to stop at those already calculated.

2. 3. P. L. Chebyshev’s formula for flat mechanisms

P.L. Chebyshev solved not only problems of synthesis of mechanisms. Many years earlier than other scientists, he deduced the famous structural formula of plane mechanisms, which only due to a misunderstanding is called the formula of Grübler, a German scientist who discovered it 14 years later than Chebyshev.

P. L. Chebyshev first proposed in 1869 a structural formula for flat mechanisms without redundant connections for lever mechanisms with rotary pairs and one degree of freedom. Currently, Chebyshev's formula has been extended to any plane mechanisms and is derived taking into account redundant connections as follows.

Let in a flat mechanism having m links (including stand), n=m-1– number of moving parts, p n– number of lower pairs and p V– number of higher pairs. If all the moving links were free bodies performing plane motion, the total number of degrees of freedom would be equal to 3n. However, each lower pair imposes two constraints on the relative motion of the links forming the pair, and each higher pair imposes one constraint, leaving 2 degrees of freedom.

The number of superimposed connections may include a certain number q P Andredundant (repeated) connections, the elimination of which does not increase the mobility of the mechanism. Consequently, the number of degrees of freedom of a flat mechanism, i.e. the number of degrees of freedom of its moving kinematic chain, relative to the stand, is determined by the following Chebyshev's formula:

Structural analysis of the mechanism

Analysis plan:

Determination of the degree of movement of the mechanism (W-?)

Division into structural groups and determination of their class and order.

Recording the formula for the structure of the mechanism.

Any mechanism (without redundant connections) consists of one (several) initial mechanisms and structural groups (Fig. 4).

Any link that has a common kinematic pair with the rack can be declared the initial link. The initial link is indicated by an arrow (Fig. 5).

The initial mechanism is understood as the combination of the selected initial link, the rack and the kinematic pair connecting them.

W=3 5-27-0=1

The degree of mobility is 1.

Each mechanism with W=1 can be considered to consist of a mechanism of the 1st class and structural groups attached to it.

The 1st class mechanism is understood as the initial link with a stand. The 1st class mechanism has W=1.

Let's consider an example of a structural analysis of a mechanism (Fig. 6).

Fig.7. Functional diagram at the level of typical mechanisms.

In Fig.6. shows a block diagram of the flat mechanism of a slotting machine, and Fig. 7. its functional diagram is at the level of standard mechanisms. The structural diagram of the mechanism, in accordance with the accepted symbols, depicts the links of the mechanism, their relative position, as well as movable and fixed connections between the links. In the diagram, the links are indicated by numbers, kinematic pairs - by capital letters. The numbers in the indices of the designation of kinematic pairs indicate the relative mobility of the links in the pair, the letters indicate the type of pair, which is determined by the type of relative movement of the links ( V - rotational, P - progressive, ts - cylindrical, VP - denotes the highest pair in which relative sliding with simultaneous rolling is possible). Scheme in Fig. 7. reflects the structure of the mechanism in the form of a serial and parallel connection of simple or standard mechanisms. In this mechanism, the rotational movement of the motor shaft φ 1 into coordinated feed movements φ 8 and dolbyak S 6 . In this case, the mechanical energy of the engine is transformed: the speed components of the energy flow decrease in magnitude, and the power components increase. The structural elements (standard mechanisms) in this scheme are interconnected by fixed connections - couplings. The diagram shows what simple mechanisms the device under study consists of, how these mechanisms are interconnected (series or parallel), how input movements are converted into output ones (in our example φ 1 V φ 8 And S 6 ).

Let us carry out a structural analysis of this mechanism. Number of moving parts of the mechanism n=8 , number of kinematic pairs p i =12 , of which for a flat mechanism single-moving p 1 =10 (rotational p 1c =8 , progressive p 1p =2 and two-movable p 2 =2 . Number of movements of the mechanism on the plane:

W pl = 3*8 - (2*10 + 1*2) = 2 = 1 + 1,

the two mobility obtained are divided into basic or specified W 0 = 1 and local W m = 1 . The main mobility determines the main function of the mechanism to transform the input movement φ 1 into two functionally interrelated φ 8 And S 6 . The local one provides an auxiliary function: in the higher cam-pusher pair, it replaces sliding friction with rolling friction.

2.4 . Mechanisms P.L. Chebysheva

But the interests of P.L. Chebyshev were not limited to considering only the theory of approximate guiding mechanisms. He dealt with other problems that were also relevant for advanced mechanical engineering.

Studying the trajectories described by individual points of the links of hinged lever mechanisms, P.L. Chebyshev stops at trajectories whose shape is symmetrical. By studying the properties of these symmetrical trajectories (crank curves), he shows that these trajectories can be used to reproduce many technically important forms of movement. In particular, he shows that it is possible to reproduce rotational motion with different directions of rotation about two axes using articulated mechanisms. One of these mechanisms, later called “paradoxical” (Fig. 8), is still the subject of surprise to all technicians and specialists. The gear ratio between the drive and driven shafts in this mechanism can vary depending on the direction of rotation of the drive shaft.

The rupture of the mechanism links has the following relationships (Fig. 9):

AC'=0.557; CC'=1.324; C 1 C=1.387;

MD=0.584; C 1 D=0.123;

The dimensions of the links C 1 D and MD are chosen so that the sum of their lengths is equal to the radius of the circle described around the trajectory of point M, and their difference is equal to the radius of the circle inscribed in this trajectory, i.e.

C 1 D+MD=R 0 and MD-C 1 D=R 1 .

A circle of radius R 0 touches the trajectory of point M at three points: M 0, M 2, M ' 2. A circle of radius R 1 also touches this trajectory at three points: M 1, M 3, M ' 3. When point M comes to positions M 0, M1, M2, M ' 2, M 3, M ' 3, then the links MD and C 1 D are extended into one line, i.e. driven link C 1 D is in limit positions. For one revolution of the crank AC’ there will be six limit positions: three external (the lengths of the links C 1 D and MD are added) and three internal (the lengths of the links C 1 D and MD are subtracted). Since the link C 1 D can leave each limit position by rotating in both one and the other direction, to determine the movement of the mechanism, the driven link C 1 D is equipped with a flywheel.

The paradox of the mechanism lies in the fact that with constant rotation of the driven link C 1 D in the direction opposite to the direction of rotation of the crank AC', it makes four revolutions per revolution of the crank. When the driven link C 1 D rotates in a direction coinciding with the direction of rotation of the crank AC’, it makes two revolutions per revolution of the crank.

P.L. Chebyshev created a number of so-called mechanisms with stops (stay). In these mechanisms, widely used in modern automotive engineering, the driven link moves intermittently. Moreover, the ratio of the rest time of the driven link to the time of its movement should change depending on the technological tasks assigned to the mechanism. P.L. Chebyshev is the first to provide a solution to the problem of designing such mechanisms. He has priority in the creation of “motion rectifier” mechanisms, which have recently been used in a number of designs of modern devices and such gears as progressive gears such as Vasant, Constantinescu and others.

In Fig.5. two diagrams of mechanisms with stands based on Chebyshev mechanisms are presented.

P.L. Chebyshev owns over 40 different mechanisms and about 80 of their modifications (Table 1.). In the history of the development of machine science, it is impossible to point out a single scientist whose work would include such a significant number of original mechanisms, and who would have such rich technical intuition.

Table 1.

List of models of mechanisms P.L. Chebysheva

	Name
	Four-link anti-rotation handle mechanism
	Scooter chair mechanism
	Six-link anti-rotation handle mechanism
	"Paradoxical" mechanism
	A mechanism that gives two swings of the driven link per one revolution of the crank
	Mechanism for converting rocking motion into rotational motion
	Bicycle mechanism
	Mechanism for converting rotational motion into translational motion with accelerated reverse motion
	Press mechanism
	Mechanism with a long stop of the driven link at the end of its stroke
	The "sorting" mechanism
	Mechanism with driven link stopping halfway
	Six-link mechanism with stops in extreme positions
	Multi-link mechanism with stops in extreme positions
	Anti-rotation handle mechanism with driven link stop
	"Stepping Mechanism" ("Stepping Machine")
	Rowing mechanism
	A mechanism that guides along an arc of a circle
	Steam engine mechanism
	Mechanism of scales
	Curvature meter
	Curvature ruler
	Variable stroke mechanism
	Arithmometer (continuous motion adding machine)
	Centrifugal regulator

2.4.1. A mechanism that produces two swings of the driven link per revolution of the crank

AC'=0.54; CC'=1.29; ω=80 o ;

MD=1.6; DF=0.81; CF=1.29; C'F=2.57.

The driven link makes two full swings per revolution of the crank: one slow and the other fast (Fig. 12) the trajectories of point M, traversed by it when the driven link DF moves from right to left, are shown in thick lines; the corresponding sections traversed by point A of the crank are also shown.

2.4.2. Mechanism for converting rocking motion into rotational motion

AB=BC=BM=1

AC"=0.545, СС'=1.325, ω=80°,

MD=1.61, FD=0.71, GF=1.33, GH=1.36,

КН=0.39, CF=1.6, С’F=2.6, KF=2.11,

The relationships between the sizes of the links are chosen so that the link KN can make a full revolution, while the link AC’ performs one full swing at a certain angle (Fig. 13). Point A makes forward and backward movements in approximately equal intervals of time. If the AC link is taken as the driving link (the “motion rectifier” mechanism), then to determine the direction of rotation when passing through the limit positions, the driven link KN must be equipped with a flywheel, which is done in the model.

2.4.3. Bicycle mechanism

The dimensions of the links have the following ratios:

AB=BC=VM=1,

AC"=0.55, SS"=1.38, ω=267°,

MK=KF=1.84, С’F=1.23, FC=1.77.

When the KF link moves from top to bottom, from one extreme position to the other, the crank AC’ makes more than half a revolution (Fig. 14). The trajectory section of point A of the crank AC’, corresponding to the reverse stroke of the AF link, is indicated by a thick line.

By attaching a second similar mechanism to the C’ axis with a crank offset relative to the AC’ crank by an angle of 180°, we get the opportunity, by pressing alternately on the driving links of both mechanisms, to rotate the driven link AC’ (the “motion rectifier” mechanism).

Judging by the nature of the model, it can be assumed that this mechanism was not intended to move a bicycle, but to be used as a foot drive.

2.4.4. Mechanism for converting rotational motion into translational motion with accelerated reverse motion

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1,

AC"=0.55, SS"=1.38,

ω=267°, γ=43.5°,

The driven link of this mechanism (slider D), moving forward, has an accelerated reverse motion. The section of the trajectory of point A, corresponding to the reverse stroke of the slider D, is shown by a thick line (Fig. 16).

2.4.5. Press mechanism

The dimensions of the mechanism links have the following ratios:

AB=BC=BM=1,A"A" = 0.198,

С"С"= 1.105, MK=0.211.

The driving link is the connecting rod A’A’’. Thus, the complex movement of the connecting rod is transformed into the translational movement of the slider T (Fig. 17).

2.4.6. Four-link anti-rotation handle mechanism

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1,

Point M describes a trajectory (Fig. 18), which differs little from a circle of radius R equal to

The direction of movement of point M along its trajectory is the opposite to the direction of movement of point A of the AC crank, and therefore this mechanism can serve as an anti-rotation handle.

2.4.7. Scooter chair mechanism

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1,

AC"=0.325, SS'=1.385.

If the connecting rod AB is made the leading link and point M is moved along its trajectory, then the crank AC' will make a full revolution, which was used by Chebyshev when he designed his scooter chair, in which each of the two wheels is driven into rotation using a mechanism (Fig. 19).

2.4.8. Six-link anti-rotation handle mechanism

The dimensions of the mechanism links have the following ratios:

AB = BC = VM = 1,

AC" = 0.54, СС'=1.33, MD=C 1 D=0.57, С 1 С=1.39,

When the crank AC' rotates, the driven link C 1 D, equipped with a flywheel, makes a full revolution in one revolution of the crank (Fig. 21). Since the rotation of the driven link C 1 D occurs in the direction opposite to the rotation of the crank AC’, this mechanism can serve as a counter-rotational handle.

2.4.9. Mechanism with a long stop of the driven link at the end of its stroke

The dimensions of the mechanism links have the following ratios:

The length of the link MD is equal to the radius of the circle to which the trajectory of point M is close in a certain section, and the position of the center F is chosen in such a way that in one of the extreme positions of the link FD point D coincides with the center of this circle (Fig. 23). As a result, the driven link FD has a stop in one of the extreme positions, the duration of which is equal to the time it takes for point M to pass the section of the trajectory close to the circle (indicated in Fig. 23 by a thick line).

For these link length ratios, the duration of the stop is approximately equal to half a revolution of the crank or the time of full swing of the FD link.

2.4.10. Sorting mechanism

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1, AC"=0.305, SS"=0.76,

MD=0.66, FD=0.8, CF=1.66, С’F=2.36

The operating principle of the device is as follows. When the DF rocker arm is in the extreme right position, grain from the hopper enters a tray mounted in the upper part of the rocker arm (Fig. 25). Since the stop of the rocker DF in this position is long and corresponds to a half-turn of the crank AC’, the grain has time to completely fill the tray. During the next half revolution of the AC crank, the DF rocker with the tray filled with grain quickly swings completely. In this case, the grains, separating from the tray, fall closer or further, depending on the size of their mass. Link NP closes the outlet of the hopper, opening it only at the moment corresponding to the stop of the rocker DF.

2.4.11. Mechanism with driven link stopping halfway

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1,

AC"=0.54, SS"=1.3,

MD=1.603, FD=0.695,

CF=1.8, C’F=2.78.

The trajectory of point M in the area indicated by the thick line differs little from the arc of a circle. The length of the link MD is taken to be equal to the radius of this circle, and the position of the center F is chosen so that in one of the middle positions of the rocker arm DF, point D comes to the center of the specified circle. As a result, with the continuous rotation of the crank AC, the rocker arm DF performs an oscillatory movement and stops in the middle of the power stroke.

The return stroke of the rocker is accelerated and without stopping.

2.4.12. Six-link mechanism with stops in extreme positions

The dimensions of the mechanism links have the following ratios:

AB=BC=BM=1, AC'=0.43, СС'=1.15,

MD=3.34, FD=0.4l, CF=1.47, C"F=2.51.

The trajectory of point M has two sections of approximately equal curvature (in Fig. 29, these sections are indicated by a thick line). The length of the link MD is taken to be equal to the radius of the circles with which the indicated sections coincide, and the position of the center F is chosen so that in the extreme positions point D comes to the centers of these circles. As a result, with continuous rotation of the crank AC, the DF link performs an oscillatory motion with stops in its extreme positions.

2.4.13. Multi-link mechanism with stops in extreme positions

The dimensions of the mechanism links have the following ratios:

AB=1, BC=4.01, TM=2.8, AF=4.44,

CE=HD=EF=PN=NQ=NO=1.31,

DL=LM=LP=QS=SR=ST=1,2,

PD=QR=1.74, DK=0.68, AT=0.5,

MO=0.28, FK=1.08, KM=1.56,

AM=2.43, AK=3.67, AO=2.4.

When the crank AB rotates, the point R of the driven link moves approximately along a circular arc with stops in extreme positions (Fig. 31).

2.4.14. Anti-rotation handle mechanism with driven link stop

The dimensions of the mechanism links have the following ratios:

AB=BC=VM=1, AC"=0.19, SS'=1.11,

MD=0.403, FD0=0.12, CF=2.05.

The trajectory of point M is close to a circle in the section corresponding to the angle of rotation of the crank AC' by 180° (in Fig. 33, this section is indicated by a thick line). The length of the link MD is equal to the radius of the circle to which the trajectory of point M is close, and the position of the center F and the length of the link FD are selected in one of the positions of the mechanism so that point D comes to the center of this circle, while the axes of the links MD and FD are extended into one line , i.e. they are in a limiting position. To ensure certainty of movement when passing through limit positions, the driven link FD is equipped with a flywheel.

When the driven link FD rotates in a direction coinciding with the direction of movement of the crank AC', it moves with a stop, the duration of which is approximately equal to the time of half a revolution of the crank. When the driven link FD rotates in the direction opposite to the direction of movement of the crank AC', the mechanism is a conventional anti-rotation handle.

2 .4.15. "Stepping Mechanism" ("Stepping Machine")

The dimensions of the mechanism links have the following ratios:

A 1 B 1 =B 1 C=B 1 M 1 =A 2 B 2 =B 2 C=B 2 M 2 =A 3 B 3 =B 3 C 1 =B 3 M 3 =

A 4 B 4 =B 4 C 1 =B 4 M4=1,

A 1 C'=A 2 C'=A 3 C' 1 =A 4 C' 1 =0.355,

SS'=C 1 C' 1 =0.785, A 2 L 4 =A 1 A 3 =C'C' 1 =0.634.

The mechanism consists of four lambda-shaped straight bars connected so that their cranks form a hinged parallelogram A 1 A 2 A 3 A 4 (link A 1 C' is rigidly connected to link A 2 C', and link A 3 C' 1 is connected to link A 4 C' 1). Points M 1 and M 4 belong to the link to which legs 1 and 4 of the mechanism are rigidly attached, points M 2 and M 3 belong to another link to which legs 2 and 3 are rigidly attached. Figure 35 shows the trajectory of the point in its movement relative to the body , i.e. link CC'C' 1 C 1 The trajectory is close to the curve that the end of the leg of a walking person or animal describes in relation to its body, namely the straight part of the trajectory of point M corresponds to the position of the end of the leg on the ground, the rest of the trajectory - movement of the end of the leg above the ground.

If from the position indicated in Fig. 35, the body SS"C' 1 C 1 is moved rectilinearly in one direction or another, then while points M 4 and M 1 remain on the straight sections of their relative trajectories, legs 1 and 4 are motionless, and legs 2 and 3 move in the direction of movement of the body. At the moment when points M 1 and M 4 must leave the straight section, points M 2 and M 3 come to the beginning of their straight section, since with the selected dimensions of the links the angle of rotation of the crank corresponding moving point M along a straight section is equal to 180°. With further movement of the body, legs 2 and 3 will remain motionless for some time, and legs 1 and 4 will begin to move in the direction of movement of the body, and thus, with continuous movement of the body, the legs of the mechanism step in a similar manner. animal's legs.

2.4.16. Rowing mechanism

The dimensions of the mechanism links have the following ratios:

AB=BC=BM=A 1 B 1 =B 1 C 1 =B 1 M 1 =1,

AC=0.297, СС'=0.765, A 1 C' 1 =0.528, C 1 C' 1 =1.21,

MM 1 =1.275, SS’=0.74, SS’=1.335, SS 1 =1.3,

When the crank AC' rotates, the oar, rigidly connected to the link MM 1 K, has a movement close to translational along the trajectories indicated for points M and K. The movement of the oar under water corresponds to the straight sections of the trajectory of points M and K (Fig. 37). Entry and exit from the water occurs almost vertically and at low speed.

2.4.17. A mechanism that guides along an arc of a circle

The dimensions of the mechanism links have the following ratios:

O B B=O B D=1,

O 1 A=1.55, AB=0.418, O 1 O8=2.18, VM=0.983,

AM=1.23, CM=2.46, CO 2 =0.526, O 1 O 2 =0.608,

O 3 O 2 =2.51, FD=1.51, O 4 F=0.92, O 4 O 3 =1.795,

With the indicated dimensions of the links, point M describes a trajectory that differs little from the arc of a circle. The radius of this circle is equal to the length of the MC link, and the center of rotation coincides with the position of point C. As a result, the O 2 C link is motionless during the entire movement of the O 4 F crank (Fig. 39).

2.4.18. Steam engine mechanism

The link sizes determined from this draft design have the following ratios:

О 1 В=1, РВ=5.38, О 1 А=0.755, AC=2.82,

EO 2 =2.08, O 2 D=1.58, DC=1.29, DE=l.54,

O 4 E=1.36, O 4 F=l.03, EF=0.78, FG=2.19,

GO 5 =2.08, MO 5 =MK==1.89, MN=0.62, AK=3.7;

NT=3.09, h=1.27, O 1 O 2 =3.38, O 1 O 5 =3.56,

O 1 O 3 =3.95, O 5 O 3 =1.94, O 5 O 2 =3.04, O 3 O 4 =2.09.

Center O 4 is mounted on lever 1 (Fig. 41), which can be installed in various positions. In this case, the trajectory of point N and, consequently, the nature of the movement of spool 2 changes. In the diagram, lever 1 is shown in the extreme left position, at which the stroke length of spool 2 is greatest.

2.4.19. Mechanism of scales

The dimensions of the links have the following ratios:

O 1 B=0.692, AB=1.5, BC=0.693, CE=0.626,

CD=0.353, DE=0.442, O 3 E=0.941, O 1 O 3 =0.692,

DF=0.98, O 2 F=0.892, O 2 O 3 =0.892, O 1 O 2 =1.42.

Link O 1 A is affected by the gravitational force of the weighed load Q, and link O 2 F is affected by the gravitational force of the counterweight P. Since the mechanism has a degree of mobility equal to two, to determine the positions of all links of the mechanism, two independent conditions should be specified. In this case, such conditions are: 1) the condition of equilibrium of the system under the influence of forces P and Q and 2) the condition of horizontality of the O 2 F link in the equilibrium position. Both conditions are satisfied at a certain position of the O 3 E link, and this position obviously changes with changes in the size of the load being weighed (Fig. 43).

The O 3 E lever is equipped with a setting device and a vernier, and a scale is mounted on the scale body. The counterweight P is made replaceable. Each counterweight value must correspond to a special scale graduation.

Flat hinge mechanisms are found everywhere in life - this is a door closer, an umbrella spoke, and a car door opening system. The work of some of them may seem surprising. For example, car windshield wipers are “windshield wipers” that quickly wipe water from the windshield in one direction or the other. Have you ever wondered how they are powered? If you look from the outside, their work looks contrary to the laws of physics: the only attachment point, the leash, presses the brush to the glass. If the motor, which we cannot see, is powerful enough to rotate such a system, then it cannot change the direction of rotation quickly enough.

Having studied the device, you can see that the motor rotates all the time in one direction, and the flat hinge mechanism - sticks connected by hinges - historically called in cars a “windshield wiper trapezoid”, converts the uniform rotation of the axis into reciprocating circular movements of the wipers

Chebyshev spent most of his professorial salary on the manufacture of the mechanisms he invented. His “plantigrade machine” is now considered the world's first walking mechanism, and it received universal approval at the Exposition universelle de 1878 in Paris. Now it is kept in the Polytechnic Museum of Moscow. The plantigrade machine did not know how to move on its own and did not know how to turn. But this was the first successful attempt to find a replacement wheel. No matter how perfect this invention of mankind, rightly revered as one of the greatest, it presupposes one essential condition - the presence of a road. In very rough terrain it is practically useless, but animals move easily there. But robotics cannot yet fully imitate their movements. Modern implementations of walking mechanisms can be seen, for example, in walking excavators or models of the Dutch kinematic sculptor Theo Jansen.

Scientific legacy of P.L. Chebyshev in the field of the theory of mechanisms contains such a wealth of ideas that paints the image of the great mathematician as a true innovator of technology.

Conclusion

World science knows few names of scientists whose creations in various branches of their science would have such a significant impact on the course of its development, as was the case with the discoveries of P.L. Chebysheva. An almost immense field of new questions, new methods for solving them stems from Chebyshev’s brilliant ideas, which arose and developed on the basis of the same philosophical thought: take nature as it is as an inevitable real fact of observation, and extract the greatest possible benefit from the observation data provided with the least expenditure of effort, “in accordance with the requirements of practice,” which, as Chebyshev himself said in his speech “On Drawing Geographic Maps,” “looks everywhere for the best, the most profitable.”

A year after Chebyshev’s death, his famous student A. M. Lyapunov wrote (in 1895): “It is unthinkable to properly assess the significance of the great scientist without a detailed analysis of his works, which is impossible without a deep study of them, and at present could not be carried out.” in any way satisfactory. The brilliant ideas scattered in the works of P. L. Chebyshev, without a doubt, not only are not exhausted in all their conclusions, but can bear proper fruit only in the future.”

Almost 120 years have passed since then, and these words remain in full force to this day, and for a long time, both scientists and practical figures around the world will draw their revelations from these “brilliant ideas of Chebyshev,” which were invariably led by me. I repeat the words of Chebyshev himself, “a general and most important thought for all practical human activities: how to manage your means to achieve the greatest possible benefit.”

From P. L. Chebyshev there comes a mathematical school that bears his name. The followers of the Chebyshev (otherwise called St. Petersburg) school of mathematics were outstanding Russian scientists: E. I. Zolotarev (1847-1878), A. A. Markov (1856-1922), A. M. Lyapunov (1857-1918), V. A Steklov (1863-1926), A. N. Krylov (1863-1945), etc. World-famous Russian mathematicians also belong to this school: S. N. Bernstein, I. M. Vinogradov, B. N. Delone. and etc.

Bibliography

S.N. Bernstein. Chebyshev, his influence on the development of mathematics. “Scientific Notes of Moscow State University,” issue 91, 1947, p. 43.

V.A. Steklov. Theory and practice in Chebyshev's research, Petrograd, 1921, p.11.

V.E. Prudnikov, P.L. Chebyshev - scientist and teacher, Uchpedgiz, 1950, p.21.

P.L. Chebyshev. Selected works. M.: Publishing House of the Academy of Sciences, 1955, 923 p.

Prudnikov V.E. Pafnuty Lvovich Chebyshev 1821-1894. L.: Nauka, 1976.

P.L. Chebyshev. Selected mathematical works. OGIZ. Gostekhizd. 1946, 189 p.

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The future great mathematician was born in 1821 to his father, a veteran of the Patriotic War, and his mother, a strict and domineering landowner typical of that time. Wanting to make their children educated people, the Chebyshev family moves from near Kaluga to Moscow, closer to the university. Today, perhaps, you will not find such stern teachers as Chebyshev had in childhood. Very little Paphnutius was taught to read and write by his iron mother, and French and arithmetic by his cousin, who was also probably not a muslin young lady. Having matured a little, the capable boy fell into the hands of a man-machine, known for his manic pedantry and harshness towards his students. The outstanding mathematician and supporter of stick discipline Platon Nikolaevich Pogorelsky firmly implanted his science in the minds of teenagers, and soon young Chebyshev began to solve complex problems faster than a squirrel nuts. By the way, the formidable Platon Nikolaevich taught mathematics to the future writer Turgenev.

A boat driven by a Chebyshev rowing mechanism. In total, at least three such waterfowl were made.

A graduate of Moscow University, he conducted his scientific activities at the University of St. Petersburg. Here he became a professor at only 29 years old, and here he created the later famous St. Petersburg School of Mathematics. While teaching mathematics, Professor Chebyshev was famous for his punctuality - he was never late for lectures, started them at a strictly appointed time and finished them exactly on the clock, even if he had to stop his story mid-sentence - there was definitely something of a robot in him.
Several of Chebyshev's students subsequently became equally famous mathematicians themselves. According to the online database “Mathematical Genealogy,” which calculates the academic pedigree of famous mathematicians, by the fall of 2013, Chebyshev, who died in 1894, had 9,609 “descendants” all over the world—people whose PhD thesis supervisors were students of his students’ students. The calculation is based on six students of Chebyshev, who defended their dissertation with him back in the 19th century. To remain in the history of mathematics as a world-famous figure, Pafnutiy Chebyshev would only need two works published by him. The first, published in 1850 in French “Memoriesurlesnombrespremiers,” took the theory of prime numbers (those that are divisible by themselves and one without a remainder) to a new level. In his 1867 work “On Average Values,” he presented calculations known today as Chebyshev’s theorem. It became one of the foundations of probability theory - the main tool of modern statistics. However, prime numbers and probability theory were drops in the ocean of Pafnutiy Lvovich’s mathematical and near-mathematical interests. Being not just a genius, but a generalist, he explored a variety of dissimilar areas of mathematics, much like Pushkin wrote frivolous poetry, poems, and historical novels with equal success.

In 1881, Chebyshev designed the world's first automatic machine for calculations, which was far ahead of all calculating machines that existed at that time. This machine, by coincidence, did not become widespread, but gave impetus to the improvement of “machine mathematics”, and then to the emergence of cybernetics.

In addition to mathematicians, mechanics and roboticists, geographers, artillerymen and... feminists consider Chebyshev to be “their people”. The first two categories pay tribute to the memory of Pafnutiy Lvovich for his contributions to improving cartography techniques and his active work on improving the range and accuracy of artillery fire. Fighters for the rights of the weaker sex remember that it was he who proposed to the physics and mathematics department of the St. Petersburg Academy to elect a woman mathematician Sofya Vasilievna Kovalevskaya as a corresponding member of the academy.

With your left foot - march in step! How the footwalker moves, see the website www.tcheb.ru

How are the mathematical works of the St. Petersburg professor and his plantigrade machine connected? Pafnuty Lvovich believed that any mathematical calculations can and should be tested in practice. So the machine designed by Chebyshev turned out to be the embodiment of two theories he developed - approximation of functions and synthesis of mechanisms. Practical mechanics was for him a continuation of his mathematical research, when numbers and symbols turn into tangible hinges and links. Chebyshev's plantigrade machine does not stand still like an idol, but walks thanks to the so-called lambda mechanisms. One of the hinges of the mechanism rotates around the axis in a circle, pushing the driven hinge, which, in turn, moves the leg with the “foot”.
One axis drives two mechanisms, that is, two legs. Accordingly, two axes - four legs. The first plantigrade machine, created by Chebyshev himself, can be seen today in the Polytechnic Museum in Moscow. A real professor can always surprise and confuse others. Chebyshev had one mechanism for this, which moved in a very mysterious way even for modern researchers. It’s called a paradoxical mechanism. Chebyshev was a true innovator, much earlier than others, he deduced the structural formula of flat mechanisms and proved the famous theorem about the existence of three-jointed four-bar mechanisms. He built a rowing mechanism that imitated the movement of boat oars, a scooter chair, and an original model of a sorting machine. In total, he created about 40 mechanisms and about 80 of their modifications, on the construction of which he spent most of his professorial salary. Without knowing it, we can still see many of the mechanisms invented by Chebyshev in modern devices today.
In addition to living heirs, Professor Chebyshev has one worthy iron descendant - the supercomputer “SKIF MSU Chebyshev” built in 2008. Today Chebyshev is one of the most powerful computing complexes in Eastern Europe. The peak performance of the supercomputer, built on 1250 quad-core processors, is 60 teraflops.

There are two objects in space named after the Russian mathematician - the Chebyshev crater on the Moon and the asteroid 2010-Chebyshev.

Chebyshev mechanism- a mechanism that converts rotational motion into motion that is close to linear.

Description

The Chebyshev mechanism was invented in the 19th century by mathematician Pafnuty Chebyshev, who conducted research on theoretical problems of kinematic mechanisms. One of these problems was the problem of converting rotational motion into something approximating linear motion.

Rectilinear movement is determined by the movement of point P - the midpoint of the link L 3, located in the middle between the two extreme coupling points of this four-bar mechanism. ( L 1 , L 2 , L 3, and L 4 are shown in the illustration). When moving along the area shown in the illustration, point P deviates from ideal linear movement. The relationships between the lengths of the links are as follows:

$L\_1:\; L\_2:\; L\_3\; =\; 2:\; 2.5:\; 1\; =\; 4:\; 5:\; 2.$

Point P lies in the middle of the link L 3. The given relations 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:

$L\_4=L\_3+\backslash sqrt(L\_2^2\; -\; L\_1^2).$

Based on the described mechanism, Chebyshev produced the world's first walking mechanism, which enjoyed great success at the World Exhibition in Paris in 1878.

see also

Other ways to convert rotational motion into approximately linear motion are the following:

• Heuken mechanism is a type of Chebyshev mechanism;

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Rotational Straight-line ...approximately Progressive Complex movement

An excerpt characterizing the Chebyshev Mechanism

-About...the wolf!...hunters! - And as if not deigning to deign the embarrassed, frightened count with further conversation, he, with all the anger he had prepared for the count, hit the sunken wet sides of the brown gelding and rushed after the hounds. The Count, as if punished, stood looking around and trying with a smile to make Semyon regret his situation. But Semyon was no longer there: he, taking a detour through the bushes, jumped the wolf from the abatis. Greyhounds also jumped over the beast from both sides. But the wolf walked through 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 dogs known to him, by the approach, distance and elevation of the voices of those arriving, he felt what was happening on the island. He knew that there were arrived (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 untoward had happened. Every second he waited for the beast to come to his side. He made thousands of different assumptions about how and from which side the animal would run and how it would poison it. Hope gave way to despair. Several times he turned to God with a prayer that the wolf would come out to him; he prayed with that passionate and conscientious feeling with which people pray in moments of great excitement, depending on an insignificant reason. “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 for this; but for the sake of God, make sure that the seasoned one comes out on me, and that Karai, in front of the “uncle” who is watching from there, slams into his throat with a death grip.” A thousand times during these half-hours, with a persistent, tense and restless gaze, Rostov looked around the edge of the forest with two sparse oak trees over an aspen underhang, and the ravine with a worn edge, and the uncle’s hat, barely visible from behind a bush to the right.
“No, this happiness will not happen,” thought Rostov, but what would it cost? Will not be! I always have misfortune, both in cards and in war, in everything.” Austerlitz and Dolokhov flashed brightly, but quickly replacing, in his imagination. “Only once in my life would I hunt down a seasoned wolf, I don’t want to do it again!” he thought, straining his hearing and vision, looking to the left and again to the right and listening to the slightest shades of the sounds of the rut. He looked again to the right and saw something running towards him across the deserted field. “No, this can’t be!” thought Rostov, sighing heavily, like a man sighs when he accomplishes something that has been long awaited by him. The greatest happiness happened - and so simply, without noise, without glitter, without commemoration. Rostov could not believe his eyes and this doubt lasted more than a second. The wolf ran forward and jumped heavily over the pothole that was on his road. It was an old beast, with a gray back and a full, reddish belly. He ran slowly, apparently convinced that no one could see him. Without breathing, Rostov looked back at the dogs. They lay and stood, not seeing the wolf and not understanding anything. Old Karai, turning his head and baring his yellow teeth, angrily looking for a flea, clicked them on his hind thighs.