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Freedom in Machinery: Volume 1, Introducing Screw Theory
This book deals with questions of freedom and constraint in machinery. It asks, for example, whether the smooth working of a machine will depend entirely upon the accuracy of its construction. As it answers such questions, it explores the geometrical interstices of the so-called screw systems at the joints in mechanism. It combines, in three dimensions directly, the twin sciences of the kinetics and the statics of rigid contacting bodies, striking deeply into the foundations of both. It introduces the idea that the kinetostatics of spatical mechanism is a valuable discipline, thus setting down for further development the beginnings of screw theory. A special feature of this book is the profusion throughout of its spectacular line drawings, which excite and assist the imagination. This is important in the general area of modern machinery and robots where an ability to think and to design such systems until the geometrical nature of the practical joints and the spatical movements of the various parts of modern machinery is properly understood, these new areas make this book an important and timely addition to the literature. This book, with its main accent upon geometry, makes its thrust at those levels of our understanding that lie below the algebra and the mere programming of modern computers. The author's direct personal style makes even the more difficult of hi geometrical and other ideas accessible to the wide range of mathematicians, research workers, and mechanical engineers who will read this book. Designers and the builders of robots will study the material carefully.
192 pages, Hardcover
First published December 20, 1984
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November 14, 2014
Jack's book (volume 1 and 2) is evidently a labor of love. It tells the story of a "Ball's creation-the theory of screws", one of whose peculiarities lies that all the results obtained in modern algebra and geometry, as distinct from analysis, seem to be directly applicable to it.
He criticized that Ball in his book about screws (1900) never speaks about the concept of mechanism except that, at the beginning of his chapter 24 where he begins to deal with his theory of screw-chain, he mentions his idea of mass-chain. Jack also pointed that by speaking in terms of impulse, mass moment of inertia, and instantaneously produced velocities both linear and angular, Ball avoided the concepts of linear and angular acceleration; he had no need for them.
Jack believes that Ball's treatment remains unique however. The geometrical theory of screws as the geometry of a particular space of five dimensions is not a mere extension of the ordinary Euclidean geometry. The six homogeneous co-ordinates of screw are, connected by an equation. That is of the form R=I, where R is a quadratic expression of the co-ordinates. All elements at infinity in our screw-spaces are given by the equation I=0, or by R=0. The absolute is thus a quadratic locus and therefore we have to deal with non-Euclidian geometry. That is why Ball never deal with practical mechanical matter of how the constraints are actually realized by links and joints in actual machinery, for that for him was also an irrelevancy.
The advantage to the theory of screws to be derived from a study of this geometry is apparent at every step. We may in our screw-space conceive curves and surfaces of from 1 to 4 dimensions, by taking one or more equations, between the co-ordinates. Of these, equations of the first degree determine the screw-complexes. But equations of the second degree, which determine quadratic complexes, or as Ball calls them screw complexes of second degree, are also constantly of use. Let U=0 be the kinetic quadratic n-system, and V=0 the potential quadratic n-system, then it follows from a well-known algebraic theorem that one set for screws of reference can in general be found which will reduce bother U and V to the sum of n squares these screws of reference are the harmonic screws. Ball concluded in Appendix II that the screw theory had been engaged in the study of Nature, they had approached the problems in the true philosophical spirit and the rewards they had obtained proved that
`Nature never did betray `The heart that truly loved her.'
He criticized that Ball in his book about screws (1900) never speaks about the concept of mechanism except that, at the beginning of his chapter 24 where he begins to deal with his theory of screw-chain, he mentions his idea of mass-chain. Jack also pointed that by speaking in terms of impulse, mass moment of inertia, and instantaneously produced velocities both linear and angular, Ball avoided the concepts of linear and angular acceleration; he had no need for them.
Jack believes that Ball's treatment remains unique however. The geometrical theory of screws as the geometry of a particular space of five dimensions is not a mere extension of the ordinary Euclidean geometry. The six homogeneous co-ordinates of screw are, connected by an equation. That is of the form R=I, where R is a quadratic expression of the co-ordinates. All elements at infinity in our screw-spaces are given by the equation I=0, or by R=0. The absolute is thus a quadratic locus and therefore we have to deal with non-Euclidian geometry. That is why Ball never deal with practical mechanical matter of how the constraints are actually realized by links and joints in actual machinery, for that for him was also an irrelevancy.
The advantage to the theory of screws to be derived from a study of this geometry is apparent at every step. We may in our screw-space conceive curves and surfaces of from 1 to 4 dimensions, by taking one or more equations, between the co-ordinates. Of these, equations of the first degree determine the screw-complexes. But equations of the second degree, which determine quadratic complexes, or as Ball calls them screw complexes of second degree, are also constantly of use. Let U=0 be the kinetic quadratic n-system, and V=0 the potential quadratic n-system, then it follows from a well-known algebraic theorem that one set for screws of reference can in general be found which will reduce bother U and V to the sum of n squares these screws of reference are the harmonic screws. Ball concluded in Appendix II that the screw theory had been engaged in the study of Nature, they had approached the problems in the true philosophical spirit and the rewards they had obtained proved that
`Nature never did betray `The heart that truly loved her.'
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