Abstract
This chapter of the book deals with gears, those developed in ancient time, as well as those to be developed in the future. The entire history of evolution of the gear art falls in three periods. These periods of evolution are referred to as (a) the pre-Eulerian period of evolution of the gear art, (b) the time when involute gearing was proposed by Leonhard Euler, and (c) the post-Eulerian period of evolution of the theory of gearing. Principal accomplishments in the field of gearing, and in the theory of gearing in particular, are identified. Each of the accomplishments is associated with the name of the principal contributor of the accomplishment, and then all the accomplishments in the theory of gearing are placed in a chronological order. In this manner, a chart that illustrates the entire evolution of the theory of gearing is constructed. The discussion in this text reveals all the gear scientists who really contributed to the scientific theory of gearing. It is helpful to eliminate numerous of other names, whose contribution is not fundamental by nature.
This chapter of the book is written in the following manner. At the beginning, a brief overview of the pre-Eulerian period of the gear art is done. Then, the fundamental accomplishments in the “scientific” theory of gearing are identified, and a name of the corresponding key contributor(s) is associated (where possible) with each of the accomplishments. As the overall number of the “fundamental” accomplishments in the scientific theory of gearing is limited, and it is not large, the overall number of the fundamental contributors to the “scientific” theory of gearing is also limited. Irrespective of many other researchers (not mentioned in this section of the book) that have also contributed a lot to the field of gearing, they cannot be regarded as the “fundamental contributors” to the “scientific” theory of gearing.
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Notes
- 1.
The involute of a circle was first proposed by Philippe de la Hire in 1696, and it was later in the eighteenth century when Leonhard Euler proposed the involute curve as a viable tooth profile.
- 2.
The term “power density” is commonly used as an equivalent to the term “power-to-weight ratio” (this concept deserves to be investigated more carefully).
- 3.
In a more general sense, that is, when non-circular gears are taken into account, use of “perfect gearing” makes possible an exact function of the pre-specified angular velocity ratio.
- 4.
The evolution of the geared mechanisms is out of the scope of this chapter of the book.
- 5.
The artifact was recovered in 1900–1901 from the Antikythera shipwreck off the Greek island of Antikythera. Its significance and complexity were not understood until decades later. Believed to have been designed and constructed by Greek scientists, the instrument has been dated either between 150 and 100 BC, or, according to a more recent view, at 205 BC. This precious example of antique genius complexity grade was so high that artefacts of a similar complexity and workmanship did not reappear for a millennium and a half, when mechanical astronomical clocks were built in Europe.
- 6.
The South-pointing chariot (invented in the fifth century BC) is another known device that contains gears. Unfortunately, only numerous nowadays designed reconstructions (not replicas) of the South-pointing chariot are known, and no original artifact remained.
- 7.
The earliest known reference to a gear was around 50 A.D.; Hero of Alexandria, through the “Book of Song,” suggests that the South-pointing chariot may have employed differential gears as early as the reign of the Zhou Dynasty (1045–256 BC) of China (Radzevich, S.P, Dudley’s Handbook of Practical Gear Design and Manufacture, 3nd ed., Boca Raton, FL: CRC Press, 2016, 629 pages.). However, no artifact of the South-pointing chariot is discovered so far. Only nowadays designed reconstructions/simulations are known. Therefore, in the meantime, the South-pointing chariot cannot be considered as a relic of a mechanism with gears.
- 8.
In 1666, R. Hooke demonstrated for The Royal Society a model of gearing that he has invented earlier. Later on the gearing of this kind Hooke described in his 1674 book Lectiones Cutlerianae. The gearing of this particular kind is nowadays known as White’s gearing. May be this is somehow associated with Mr. Christopher White of London who manufactured a microscope for R. Hooke.
- 9.
Albrecht Dürer (May 21, 1471–6 April 6, 1528), a German painter, printmaker, mathematician, engraver, and theorist
- 10.
Girard Desargues (February 21, 1591–September 1661) was a French mathematician and engineer.
- 11.
Philippe de la Hire (March 18, 1640–April 21, 1718) was a French physicist, astronomer, mathematician, and engineer.
- 12.
Charles Étienne Louis Camus (August 25, 1699–February 2, May 4, 1768) was a French mathematician and mechanician
- 13.
The line of action, KBC, cannot be a curve, as a force acts only along a straight line, that is, along a straight line of action, and not along a curve.
- 14.
It is instructive to note here that the schematic shown in Fig. 6.3 is a kind of mistake because of the following reasons. First, the path of contact is an envelope to consecutive positions of the instant line of action. Therefore, it is not permissible that the line of action, BC, intersects the path of contact, KBC. The path of contact must be in tangency with the line of action, BC. Second, when numerous instant lines of contact are through the pitch point C, then no enveloping curve (i.e., no path of contact) can be constructed. A few more reasons for infeasibility of gearing shown in Fig. 6.3 are to be mentioned.
- 15.
Leonhard Euler (April 15, 1707–September 18, 1783) was a pioneering Swiss mathematician and physicist
- 16.
Invention of the involute tooth profile, which best fits the practical needs of the industry, is commonly credited to Leonhard Euler (1707–1783).
- 17.
The consequences from the Euler-Savary formula (the involute tooth profile, and the conjugate action law) are important to the theory of gearing, while the formula itself is of less importance.
- 18.
Felix Savary (October 4, 1797–July, 15, 1841) was a French mathematician and mechanician
- 19.
Félix Savary was the first to derive the Euler-Savary formula in its modern form. Savary’s proof can be found in: Leçons et cours autographiés, Notes sur les machines, par le professeur F. Savary, Ecole Polytechnique, 1835–1836 (unpublished lecture notes; available in the Bibliothéque Nationale in Paris).
- 20.
- 21.
Reverend Robert Willis (February 27, 1800–February 28, 1875), an English academic, was a professor at Cambridge.
- 22.
Théodore Olivier (January 21, 1793–August 5, 1853), a French mathematician and engineer
- 23.
It is likely Dr. Fraifeld [23] is among those most affected (influenced) with the two “Olivier principles.” Generating (hobbing) of gears for “Novikov gearing” is another example where ignorance of the condition of conjugacy resulted in insufficient accuracy of the machined gears.
- 24.
US Patent No. 5.647, Rack and Pinion, Amzi C. Semple, June 27, 1848
- 25.
Thomas Tredgold (August 22, 1788–January 28, 1829), an English engineer and author
- 26.
Chaim I. Gochman (1851–1916), a Russian mechanician (Novorossiysk University, Odessa, now in Ukraine)
- 27.
For details, the interested reader is referred to the paper by the author: Radzevich, S.P., “Briefly on the Kinematic Method and on the History of the Equation of Contact in the Form of n ⋅ V = 0,” In: Theory of Mechanisms and Machines, 2010, No. 1. Vol. 15, pp. 42–51. http://tmm.spbstu.ru
- 28.
It could happen that the equation of contact, n ⋅ V = 0, can be found out even in earlier (before 1948) publications by Professor V.A. Shishkov – in his earlier papers, PhD thesis, and so forth.
- 29.
Conjugate tooth profiles/surfaces are also known as “reversibly-enveloping” profiles/surfaces (or just as Re−profiles/surfaces, for simplicity) [19].
- 30.
It is a right point to mention here that the author failed trying to identify the name of a gear researcher who should be credited with this fundamental requirement in the theory of gearing.
- 31.
George Barnard Grant (December 21, 1849–August 16, 1917) is considered one of the founders of gear-cutting industry in the USA (Grant established a gear-cutting machine shop in Charlestown, Massachusetts. When this business expanded, he moved the workshop to Boston, expanded it, and named it the Grant Gear Works. From this extremely successful establishment evolved the Philadelphia Gear Works and the Cleveland Gear Works. George Grant even wrote several very successful books on the subject, for example, A Treatise on Gear Wheels; A Handbook on the Teeth of Gears, Their Curves, Properties and Practical Construction, and so forth).
- 32.
Mikhail L. Novikov (March 25, 1915–August 19, 1957), a famous Soviet gear researcher
- 33.
The first pair of “Novikov gearing” made of aluminum alloy (a pre-prototype) was cut on April 25, 1954, by a disk-type mill cutter. For testing, 15 gear pairs were machined in the summer of 1954 by the disk-type mill cutter. Hobs for cutting gears for “Novikov gearing” were proposed later on by Professor V.N. Kudr’avtsev (as early as in 1956) – this is a huge mistake committed by Professor V.N. Kudr’avtsev to cut gears for “Novikov gearing” by hobs.
- 34.
Vladimir A. Gavrilenko (June 21, 1899–June 6, 1977), Doctor (Engineering) Sciences and Professor of Mechanical Engineering (Bauman State Technical University, Moscow, Russia)
- 35.
Jack Raymond Phillips (July 18, 1923–January 11, 2009), a famous Australian gear expert (mechanician)
- 36.
Walton Clarence Musser (April 5, 1909–June 8, 1998), a famous American inventor; he is the inventor of the “harmonic drive” (1957).
- 37.
Per the author’s opinion, G. Grant did not realize the importance of his invention. In the time of Grant, the industry was fulfilled with the available on the market approximate gears; no interest to precision (and more costly) bevel gears was indicated by the industry at that time.
- 38.
For more in detail discussion on manufacture of gears for approximate gearing, the interested reader may wish to go to Chap. 1 “Gears: Brief Notes on the History of Methods of Machining Gears and of Design of Gear Cutting Tools” in the book: Radzevich, S.P., Gear Cutting Tools: Science and Engineering, CRC press, Boca Raton, Florida, 2017, 606 pages.
- 39.
Friedrich Wilhelm Lorenz (1842–1924), Doctor of Engineering, inventor, and founder of the Lorenz Company
- 40.
Samuel I. Cone (1842–1924), a civilian machinist and draftsman, an American inventor of double-enveloping worm gearing
- 41.
William Gleason (1836–1922), founder of The Gleason Works, Rochester, NY
- 42.
Nikola John Trbojevich (May 21, 1886–December 2, 1973), also known as Nicholas J. Terbo, a world-known research engineer, mathematician, and inventor, held the basic patent for the Hypoid Gear.
- 43.
Ernest Wildhaber (1893–1979), Doctor of Engineering, h.c., Inventor, and consultant for The Gleason Works
- 44.
It needs to be stressed here that involute of a circle itself was known long before the invention of involute gearing by L. Euler.
- 45.
It is a right point to stress here that the “dead end” in the diagram in Fig. 6.22 means that no “perfect” Ia− and Ca−gearing are possible; no “correct” tooth flank modification in Pa−gearing is possible; and trial and error method is dominated.
- 46.
Theory of gearing can be viewed as a kind of “road map” that helps the user traveling from one point (location) to another point (location) in a most efficient way.
- 47.
After about 40 (!) PhD theses and 5 (!) Dr. Sci theses are defended by these people, how is it permissible to ask a question: “What do we know about spiroid gearing”? What did you do all this time?
- 48.
Amazingly, but this stupid “gearing” is supported by two doctors of sciences (Dr. Scherbakov, N.R., the chairperson of “Geometry” department, and Dr. Bubenchikov, A.M., the chairperson of “Theoretical Mechanics” department, both of Tomsk State University, Russia) who are granted with scientific degree of Dr.Sci. in mathematics and physics.
- 49.
Except of the contributions by L. Euler, the contributions by other members of the Hall of Fame at the Gear Research Center (The University of Illinois at Chicago) are out of the scope of the scientific theory of gearing, and, thus, are not discussed here.
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Further Readings
Babichev, D. T., Lagutin, S. A., & Barmina, N. A. (2016). Overview of the Works of the Russian School of Theory of and the Geometry of Gearing. Part 1. Origins of the Theory of Gearing, and its Heyday Time in 1935–1975. Theory of Mechanisms and Machines, 14(№3(31)), 101–134.
Babichev, D. T., Lagutin, S. A., & Barmina, N. A. (2017). Overview of the Works of the Russian School of Theory of and the Geometry of Gearing. Part 2. Development of the Classical Theory of Gearing and Establishment of the Theory of Real Gearing in 1976–2000. Theory of Mechanisms and Machines, 15(№3(35)), 86–119.
Babichev, D. T., Lagutin, S. A., & Barmina, N. A. (2020). Russian School of the Theory and Geometry of Gearing. Part 2. Development of the Classical Theory of Gearing and Establishment of the Theory of Real Gearing in 1976–2000, pages 1–46. In V. I. Goldfarb, E. Trubachov, & N. Barmina (Eds.), New Approaches to Gear Design and Production, (Mechanisms and Machine Science, Book 81) (p. 529). Springer. ISBN-13: 978–3030349448, ISBN-10: 3030349446.
Babichev, D. T., & Volkov, A. E. (2015). History of Evolution of the Theory of Gearing. Journal of Scientific and Technological Development, № 5(93), 25–42.
Crosher, W. P. (2014). A gear chronology: Significant events and dates affecting gear development (p. 260). Xlibris Corporation.
da Vinci, L. (1974). The Madrid codices, volume 1, 1493, facsimile edition of codex Madrid 1, original Spanish title: Tratado de Estatica y Mechanica en Italiano. McGraw Hill Book Company.
de la Hire, P. (1694). Mémoires de Mathématique et de Physique, Impr. Paris: Royale.
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Dong, X. (1989). Theoretical Foundation of Gear Meshing. Beijing: China Machine Press.
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Radzevich, S.P. (2022). A Brief Overview of the Evolution of the Scientific Theory of Gearing. In: Radzevich, S.P. (eds) Recent Advances in Gearing. Springer, Cham. https://doi.org/10.1007/978-3-030-64638-7_6
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