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Historical Contributions to Screw Theory

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Historical Contributions to Screw Theory Sir Robert Stawell Ball (1840-1913) • A Treatise on the Theory of Historical Contributions to Screws (1900, 1998), (1876) – most comprehensive text on screw Screw Theory theory Harvey Lipkin • Chief proponent of screw geometry Summer Screws 2009 August 22-30 • Screw theory was his avocation University of Genoa, Italy not his principal research Sir Robert Stawell Ball (1840-1913) Sir Robert Stawell Ball (1840-1913) • Royal Astronomer of Ireland • Screw kinematics and dynamics. – Also held by W.R. Hamilton • Dynamics of rigid bodies at rest (linear geometry of dynamics) • Lowndean of Astronomy and • Uniquely employed screw coordinates Geometry at Cambridge – not currently used University • No known successor to further screw • Wrote many books on theory astronomy for the public – Dynamicist E.T. Whittaker was a student • Well-known public speaker Sir Robert Stawell Ball (1840-1913) Louis Poinsot (1777 – 1859) • Ecole Polytechnique • Died before finishing • Introduced geometric autobiography/biography – screw mechanics related materials not finished • (1806) • Ball 2000 Symposium at – concept of a couple Cambridge University – concept of a wrench: any system • Treatise republished (1998) of forces can be reduced to a single force and a couple in a • Extensive annotated bibliography plane normal to the force “of principal works known to me” f o no() – Ball: a wrench on a screw Michel Chasles (1793 – 1880) Sir William Rowan Hamilton (1805 – 1865) • Royal Astronomer of Ireland • Ecole Polytechnique (predates Ball) • Projective geometry • Famous dynamicist • (1830) concept of a twist – Hamilton’s principal – a rotation along with a parallel • Invented quaternions translation ω • (1830) Discoverer of the o vo() cylindroid, the surface that describes a two-system of screws – finite and infinitesimal twists – Ball: a twist on a screw Arthur Cayley (1821-1895) Hermann Grassmann (1809-1877) • “High priest” of British • (1844) Theory of Extension mathematicians – Basis of vector algebra and other • Geometry of n dimensions branches of mathematics – General and abstract geometry • Linear algebra (w/Sylvester) – Little noticed • (1859) Projective geometry most • (1862) Republished general – used coordinates and determinant • (1860) Six coordinates of a line principles (predates Plücker) – Special cases relate to screw theory • Lawyer for 14 years, then at Cambridge University Julius Plücker (1801-1868) Felix Klein (1849-1925) • (1831) Projective geometry • Student of Plücker – (1868) finished Plücker’s book on line – Developed principal of duality geometry analytical • Defined geometry in terms of groups and – Use of homogeneous coordinates invariants • (1865-1869) Plücker coordinates • Completed Cayley’s unification of projective and metrical geometries – Ray and axis coordinates of a line • Screws as points in P5 using determinants – Lines lie on a quadric surface • Special case of Grassmann’s theory • Klein coordinates (screws) – Dynames (twists and wrenches) • (1878) edited Grassmann’s collected – Linear complex (five-system) works Felix Klein (1849-1925) William Kingdon Clifford (1845-1879) • Influential international leader in • (1872) Biquaternions mathematics and mathematical – Generalized quaternions to finite spatial education displacements – (1908) Elementary Mathematics from – Introduced “dual complex numbers” • a + ωb where ω2 = 0 (scalar) an Advanced Standpoint: Geometry • a + ωb where ω2 = 0 (motor/dual vector) • Plücker coordinates, line complex, – Investigated elliptic geometry Grassmann principle, Erlangen • Clifford algebras program, Cayley’s principle – Generalization of Grassmann algebras th • Best text on 19 century geometry • Died age 33 Ball – Clifford (1870s) Ball – Clifford – Klein (1870s) • “In the early days of the ‘Theory of Screws’ Ball • “at a meeting of the British Association in the 'seventies, there turned up a young geometer from the University of had made the acquaintance of W. K. Clifford at Erlangen, Felix Klein, already a leader in the German meetings of the British Association. They were mathematical world. Klein told them about Plücker's linear drawn together by a common interest in complex, and certain recent developments in the fascinating geometrical forms, including non-Euclidean field of geometrical relations which it involved, a field in which Klein had first shown his own genius. Ball described geometry; and both had a play of humour and frolic how he and Clifford captured Klein after the meeting, and which made them for some years main upholders of sat up half the night exchanging ideas, the interview the lighter side of the activities of that scientific culminating with the impatient and admiring complaint that there was positively nothing they could tell him that Klein body.” did not seem to know about already.“ Sir Joseph Larmor from Ball’s autobiography Sir Joseph Larmor from Ball’s autobiography Sophus Lie (1842-1899) Aleksandr P. Kotelnikov (1865 – 1944) • (1872) Isomorphism between Plücker coordinates and sphere • (1895) Developed screw theory coordinates using biquaternions • (1888) Lie groups and Lie • Principle of transference algebras between spherical and spatial – Theory of continuous groups and geometry (using biquaternions / tangent fields dual numbers) – Used to model finite and infinitesimal twists • Influenced later Russian – Screw cross product is a Lie research bracket • College roommate of Klein Eduard Study (1862 – 1930) Richard von Mises (1883 – 1953) • (1903) Developed screw theory • Important contributions in the area using biquaternions of mechanics • Principle of transference – E.g. von Mises stress, aerodynamics between spherical and spatial • (1924) Motor Calculus geometry (using biquaternions / – Very modern treatment with applications to mechanics dual numbers) – Influential on German research • Influenced later German • Married to Hilda Geiringer research – (1942) Delivered lectures on Motor Calculus at Brown University Kenneth H. Hunt (1920-2002) and Louis Brand and Heinrich Guggenheimer Jack Phillips (1923-2009) •Brand • (1962) “Rediscovered” the cylindroid and – (1947) Vector and Tensor Analysis work of Ball • Has chapter on motor algebra • Revived screw theory in 1960s • Develops motor/screw/spatial derivative – (with fundamental misprint latter corrected) • Applied to spatial mechanisms – Advisor (Bôcher) was student of Klein • Hunt • Guggenheimer – (1979) Kinematic Geometry of Mechanisms – (1963) Differential Geometry – (2004) Robots and Screw Theory (with J. • Treatment includes the use of dual-numbers Davidson) • Advisor of his advisor was a student of Klein • Phillips – (1984) Freedom in Machinery Vol.1 – (1990) Freedom in Machinery Vol. 2 – Based on synthetic (i.e. descriptive methods) F. M. Dimentberg (1905?-1999?) Ferdinand Freudenstein (1926)-(2006) • Analysis of spatial mechanisms • “Father of modern kinematics in the US” • Used a hybrid of motor algebra and • Analytical methods of mechanism design Plücker coordinates • (1960s) Study of spatial mechanisms – Applied work of Dimentberg • Screw properties of inertia and • Revitalized screw theory in US along stiffness with many of his students including Yang • (1965) English translation of The and Roth Screw Calculus and Its Applications – Popularized works of Brand, Klein, Ball, in Mechanics Guggenheimer, Dimentberg, and others.
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