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A general theory of the kinematics and geometry of gears in three dimensions
584
,,uch concepts its a linear space, norm. (Frechet) differential, differentiable manifold, etc.. it is not only possible but is in fact clearer and ea-,ier to use coordinate fi-ee notation. This is especially true ~,, hen writing equation-, involving vector and tensor fields. Finall,,. perbons other than mathematicians ma,, find thi,, book a bit too mathematicall.,, ",ophisticated. This. however, m a y be an indictment again,,t our manner of training scienti.,,ts rather than a criticism or" the text. For, the revie,,ver does not ~,ee ho,a one can do justice to the subject at hand imd not introduce many of the advanced concept,, (functional. differentiable manit'old, t.ie group, etc.) that Dr. Edelen doe,,, l e t it suffice to sa,, that it person attempting to read this book -,hould alread,, posse<,, enough mathematical maturity that Chapter three is essentiall,, a revie,a. DONAI.D
E. C A T L I N
(/tEcersio" +{/':l,la.~saciu~serr,s' A m/zer.st 3.1as,~. 0 / 002 U.S.,4.
A Dyson: A General Theory of the Kinematics and Geometry of Gears in three Dimensions Clarendon Press, Oxford, 1969. The following important problems of gearing are discussed in the book: (a) Generating of conjugate gears with line and point contact: (b) investigation of conjugate gear surfaces and their meshing conditions in the point contact locality: (c) design of conjugate gears for optimum working conditions. T,,vo methods of conjugate gears generating are discussed: (a) when two gears are generated by c o m m o n generating surface: (b) when two gears are generated by two rigidly tied up surfaces, that are in line contact with one another. Generating the gears by the first method we will have line contact between generated surfaces and in some cases point contact; generating the gears by the second method we will al,,vays have point contact between the generated surfaces. Equations are established, connecting main curvatures and their directions, when gears are in point and in line contact, the direction of motion of point of contact when gears are in point contact. The form of line of contact and the surface of action are determined for the gears, that are in line contact. The connection between the form of line of contact with the following working conditions is investigated: (a) occurrence of oil film: (b) flash temperature. The conditions are determined For the angular velocity ratio to remain constant. The advantages of the gears with point contact are justly underlined in the book, for such gears are less sensitive to the deflections of teeth and defects of manut'acturing and assembly. In the appendices to the book generation of conjugate gears mounted on parallel axes with the circular and elliptical basic racks is described. The racks have circt, lar and elliptical profiles in the cross sections orthogonal to the gears axes. In the appendices equations are given for determination of the curvatures of the surfaces, described by Cartesian and parametric equations, the E u l e r - S a v a r y equation etc. For solution of problems the classical methods of differential geometry are used in the book. For to study the structure of the surface the method of lines is used. Such
585 method was used in the Soviet Union by G. 1. Scheveleva (Look for the bibliography in the "'Theory of tooth gearings" by F. L. Litvin, publishing house "'Nauka'" 1968). The style of the book is concise, strict and distinct: it seems to me that its publishing is of undoubtedly interest for the specialists in gearings. The reviewing of the book is the occasion for evaluation of the theory of gearings state. We may ascertain that the theory of gearings becomes an independent branch of Mechanics, and its development is accompanied by elaboration of its own methods and simplification of the methods of differential geometry. Kinematic methods developed in the works of American and Soviet scientists turned out to be especially fruitful (look for the bibliography in the F. L. Litvin's book). The elaboration of approximate gears with the improved working conditions is of a great significance. As an example of such gears we may take particularly Gleason gears. The securing of favourable conditions of meshing in the point contact locality is only the first stage of synthesis of such a gearing. To solve the problem means to work out the mathematical model of all the process of meshing and optimization of meshing conditions: (a) the minimizing of fluctuations of angular velocity ratio (b) the securing of necessary form and dimensions of the spot of contact, that is the totality of the ellipses of contact. F. L. L I T V I N Institute for Precision Mechanics and Optics Leningrad Center U.S.S.R.
E. Meyer: Mechanical Seals (translated from the German): American Elsevier Publishing Co. Inc., New York, 1970. |N THE early sixties there were virtually no books on seals; therefore, this monograph represented a valuable contribution to seal technology. Also in the sixties, the need for better understanding of seal design and performance was emphasized, with the result that considerably more-valuable information is available t o d a y - w h i c h the author completely neglects. This new body of information is presented in such publications as: (l) Proceedings of the International Conference on Fluid Sealing (196 l, 1964, 1967 and 1969), published by the British Hydromechanics Research Association in England; (2) Bearing and Seal Design in Nuclear Power Machinery, ASME book dated June i 967; and (3) Journal of Lubrication Technology, April 1968 and October 1969. The chapter entitled "Fundamentals", unfortunately, is not that; which makes the book lose much of its permanent value. There is very little comparison between theory and practice- which reduces the broad applicability of the material presented. There is virtually no discussion of hydrodynamic, elasto-hydrodynamic or thermoelastohydrodynamic lubrication which takes place in a majority of mechanical seals used today. The author discusses boundary lubrication primarily, which applies only to very heavily loaded s e a l s - w h i c h generally fail by the mechanism of wear. In the chapter on "Wear", there are valuable tables of physical and mechanical properties of sliding materials. Unfortunately, the author does not clearly indicate what material criteria should be used in seal design.
,,uch concepts its a linear space, norm. (Frechet) differential, differentiable manifold, etc.. it is not only possible but is in fact clearer and ea-,ier to use coordinate fi-ee notation. This is especially true ~,, hen writing equation-, involving vector and tensor fields. Finall,,. perbons other than mathematicians ma,, find thi,, book a bit too mathematicall.,, ",ophisticated. This. however, m a y be an indictment again,,t our manner of training scienti.,,ts rather than a criticism or" the text. For, the revie,,ver does not ~,ee ho,a one can do justice to the subject at hand imd not introduce many of the advanced concept,, (functional. differentiable manit'old, t.ie group, etc.) that Dr. Edelen doe,,, l e t it suffice to sa,, that it person attempting to read this book -,hould alread,, posse<,, enough mathematical maturity that Chapter three is essentiall,, a revie,a. DONAI.D
E. C A T L I N
(/tEcersio" +{/':l,la.~saciu~serr,s' A m/zer.st 3.1as,~. 0 / 002 U.S.,4.
A Dyson: A General Theory of the Kinematics and Geometry of Gears in three Dimensions Clarendon Press, Oxford, 1969. The following important problems of gearing are discussed in the book: (a) Generating of conjugate gears with line and point contact: (b) investigation of conjugate gear surfaces and their meshing conditions in the point contact locality: (c) design of conjugate gears for optimum working conditions. T,,vo methods of conjugate gears generating are discussed: (a) when two gears are generated by c o m m o n generating surface: (b) when two gears are generated by two rigidly tied up surfaces, that are in line contact with one another. Generating the gears by the first method we will have line contact between generated surfaces and in some cases point contact; generating the gears by the second method we will al,,vays have point contact between the generated surfaces. Equations are established, connecting main curvatures and their directions, when gears are in point and in line contact, the direction of motion of point of contact when gears are in point contact. The form of line of contact and the surface of action are determined for the gears, that are in line contact. The connection between the form of line of contact with the following working conditions is investigated: (a) occurrence of oil film: (b) flash temperature. The conditions are determined For the angular velocity ratio to remain constant. The advantages of the gears with point contact are justly underlined in the book, for such gears are less sensitive to the deflections of teeth and defects of manut'acturing and assembly. In the appendices to the book generation of conjugate gears mounted on parallel axes with the circular and elliptical basic racks is described. The racks have circt, lar and elliptical profiles in the cross sections orthogonal to the gears axes. In the appendices equations are given for determination of the curvatures of the surfaces, described by Cartesian and parametric equations, the E u l e r - S a v a r y equation etc. For solution of problems the classical methods of differential geometry are used in the book. For to study the structure of the surface the method of lines is used. Such
585 method was used in the Soviet Union by G. 1. Scheveleva (Look for the bibliography in the "'Theory of tooth gearings" by F. L. Litvin, publishing house "'Nauka'" 1968). The style of the book is concise, strict and distinct: it seems to me that its publishing is of undoubtedly interest for the specialists in gearings. The reviewing of the book is the occasion for evaluation of the theory of gearings state. We may ascertain that the theory of gearings becomes an independent branch of Mechanics, and its development is accompanied by elaboration of its own methods and simplification of the methods of differential geometry. Kinematic methods developed in the works of American and Soviet scientists turned out to be especially fruitful (look for the bibliography in the F. L. Litvin's book). The elaboration of approximate gears with the improved working conditions is of a great significance. As an example of such gears we may take particularly Gleason gears. The securing of favourable conditions of meshing in the point contact locality is only the first stage of synthesis of such a gearing. To solve the problem means to work out the mathematical model of all the process of meshing and optimization of meshing conditions: (a) the minimizing of fluctuations of angular velocity ratio (b) the securing of necessary form and dimensions of the spot of contact, that is the totality of the ellipses of contact. F. L. L I T V I N Institute for Precision Mechanics and Optics Leningrad Center U.S.S.R.
E. Meyer: Mechanical Seals (translated from the German): American Elsevier Publishing Co. Inc., New York, 1970. |N THE early sixties there were virtually no books on seals; therefore, this monograph represented a valuable contribution to seal technology. Also in the sixties, the need for better understanding of seal design and performance was emphasized, with the result that considerably more-valuable information is available t o d a y - w h i c h the author completely neglects. This new body of information is presented in such publications as: (l) Proceedings of the International Conference on Fluid Sealing (196 l, 1964, 1967 and 1969), published by the British Hydromechanics Research Association in England; (2) Bearing and Seal Design in Nuclear Power Machinery, ASME book dated June i 967; and (3) Journal of Lubrication Technology, April 1968 and October 1969. The chapter entitled "Fundamentals", unfortunately, is not that; which makes the book lose much of its permanent value. There is very little comparison between theory and practice- which reduces the broad applicability of the material presented. There is virtually no discussion of hydrodynamic, elasto-hydrodynamic or thermoelastohydrodynamic lubrication which takes place in a majority of mechanical seals used today. The author discusses boundary lubrication primarily, which applies only to very heavily loaded s e a l s - w h i c h generally fail by the mechanism of wear. In the chapter on "Wear", there are valuable tables of physical and mechanical properties of sliding materials. Unfortunately, the author does not clearly indicate what material criteria should be used in seal design.