Nowacki H. Leonhard Euler and the theory of ships (794398), страница 9
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This analogyis founded on the fact that both system types to the first approximation (smallamplitudes) constitute linear oscillators with isochronous periodic oscillations, as theequations of motion will readily demonstrate. If we follow Euler’s derivation for thependulum and for the rolling ship (small roll angle φ) for a free oscillation, thefollowing comparison can be drawn:Pendulum (Mass m, center of gravity distance from center of rotation s):Equation of motion: Θ (d2φ/dt2) + m g s φ = 0Equivalent pendulum length: lEQU = Θ/(m s)Natural period:T=2πl EQU gRolling Ship (Mass m, metacentric radius GM):Equation of motion: Θ (d2φ/dt2) + m g GM φ = 0Equivalent pendulum length: lEQU = Θ/(m GM)Natural period:T = 2 π l EQU g = 2 π " (!GM )where Θ = mass moment of inertia in rolling (φ)The analogies are clearly visible, especially between s and GM.Corresponding results were obtained by Johann Bernoulli for heaving and by Euler andDaniel Bernoulli for pitching, by Daniel Bernoulli also for coupled rolling and heaving[59].In design it was the purpose to avoid great accelerations at resonance, thus to reduce thenatural frequencies and to increase the natural periods.
In rolling, e.g., where GM mustnot be chosen too small, Euler (like others) recommended to increase the mass momentof inertia Θ by shifting any movable masses inside the ship as far away from the centerof gravity as possible. A plausible idea, but only practicable within narrow limits.It should be noted with interest, too, that Euler in a later treatise [23] on rolling andpitching almost in passing ((§16-§19) also mentions how to determine the internal loadsof the ship, e.g., at midship section.
In his ingenious way Euler takes a planar verticalsection through the hull girder and determines the longitudinal bending moment as aninternal load in this cross section. By this influence he also explains the deflection of thehull in longitudinal bending, the much feared “hogging”. The article further elaborateson how dynamic loads must be added to to the static loading. This to my knowledge isthe first historical entry point into those methods which later developed intolongitudinal strength calculations for ships and became indispensable in dimensioningthe structure of the hull.The treatment of ship oscillations by Euler and his contemporaries however still hadsignificant gaps and limitations:-Neglect of hydrodynamic mass and damping forces,- 25 ---Lack of load assumptions for excitation forces and moments, especially by theseaway, hence lack of data for forced oscillations,Simplification in the choice of oscillation axes through the center of gravity asbeing parallel and orthogonal to the waterplane in place of the principal axes ofinertia,Limitation to small amplitudes, linearization,Absence of statistical methods for frequency and extreme value analysis.Euler was aware of the majority of these limitations, as his cautious premises anddisclaimers usually indicate.
Nevertheless we must recognize and pay our tribute to theuseful knowledge already achieved in Euler’s era, both in scientific substance and inqualitative practical insights.4.ConclusionsBy consistent application of the first principles of mechanics and fluid mechanics,which Euler in part had created or extended himself, he was able to base the new,application oriented scientific discipline of ship theory on a firm foundation and therebyto help lay the ground for modern ship theory. He left his mark on the structure of thisfield. Many of his results are still valid and of lasting value.
The followingachievements deserve to be singled out as especially noteworthy:o The foundation of criteria and calculation methods for the hydrostatic stability ofships, derived by integration of the pressure distribution in the fluid at rest, acting on theship slightly displaced from equlibrium. The application of stability criteria already atthe design stage as a starting point for more systematic analysis of the safety of ships.o Initially application of Newton’s impact theory of resistance, which yielded falsepredictions, yet combining correct system dynamics with false force coefficients. Laterafter intensive efforts discarding of the misleading impact theory and creation of thepromising field theory.o Contributions to the laws of ship propulsion by sailing, rowing, paddle wheel,screw propeller and jet propulsion.
Correct application of the momentum theorem toexplaining the principles of propulsion, unfortunately with wrong force coefficients.o Fundamental studies on system dynamics of the maneuvering vessel with floweffects on hull, sail and rudder, also solution of the equations of motion.o Contributions to ship oscillations by deriving the natural periods for rolling,heaving and pitching based on inertia and hydrostatic restoring forces.In all these activities we owe Euler our gratitude not only for his important insights inthe theory of ships, but also for what Truesdell [3] already praised as the specialcharacteristic of style in all of Euler’s work:“First principles, generality, order and above all clarity”.It took a major span of time before Eulers insights were understood by the engineersand found acceptance in practice.
But even in his lifetime there were a few enlightenedcontemporaries and practitioners who recognized and appreciated the value oftheoretically well founded knowledge. The eminent Swedish naval constructor F.H. afChapman [56] may be quoted here as a key witness who had comprehensive practical- 26 -experience in shipbuilding, but was also familiar with the literature of his era, hencealso with Bouguer’s and Euler’s treatises, which he held in high esteem.
He stated:“Without a good theory design is only a game of hazard!”AcknowledgmentsThe author expresses his sincere gratitude to the Max Planck Institute of the History ofScience in Berlin, where he has served as a Visiting Scholar since 2001, for alwaysgenerous support and advice. The present study was supported in particular by theavailable rich reference material. Further he acknowledges with thanks the usefulinformation provided by Jörg Blaurock, Hamburg, on the history of the paddle wheel.He is also grateful to his friends Walter Debler and Harry Benford for carefulproofreading and good advice.A condensed version of this article was presented in the Captain Ralph R.
and FlorencePeachman Lecture at the University of Michigan in Ann Arbor on April 16, 2007. Thisoccasion provided an additional incentive for preparing these notes in commemorationof Leonhard Euler’s tercentenary and of his contributions to ship theory..References1. Leonhard Euler: “Scientia Navalis seu Tractatus de Construendis ac DirigendisNavibus”, 2 vols., St. Petersburg, 1749. Reprinted in Euler’s Opera Omnia [6], Series II,vols.
18 and 19, Zurich and Basel, 1967 and 1972.2. Pierre Bouguer: “Traité du Navire, de sa Construction et de ses Mouvemens”,Jombert, Paris, 1746.3. Truesdell, Clifford A.: “Rational Fluid Mechanics, 1687-1765”, in Leonhardi EuleriOpera Omnia [6] (“Complete Works of Leonhard Euler”), issued by the EulerCommission of the Swiss Academy of Natural Sciences, Series II, vol. 12, pp. VIICXXV, Lausanne, 1954.4. Eckert, Michael: “Euler and the Fountains of Sanssouci”, Archive for History ofExact Sciences, vol. 56, no.6, Springer, Berlin/Heidelberg, November 2002, pp. 451468.5.
Fellmann, E.A.,: “Leonhard Euler”, translated from German by Erika Gautschi andWalter Gautschi, Birkhäuser Verlag, Basel-Boston-Berlin, 2007.6. Euler, Leonhard: “Leonhardi Euleri Opera Omnia” (“Collected Works of LeonhardEuler”), issued by the Euler-Commission of the Swiss Academy of Natural Sciences, infour series, some volumes still in preparation, published in Leipzig and Berlin since1911, in Basel since 1975.7.
Habicht, Walter: “Leonhard Eulers Schiffstheorie, Einleitung und Kommentar zu denBänden 18 und 19 der zweiten Serie” (“Leonhard Euler’s Ship Theory, Introduction andCommentary to vols. 18 and 19 of Series II”), in [6], Series II, vol. 21,“Commentationes Mechanicae et Astronomicae ad Scientiam Navalem Pertinentes”,vol. II, W. Habicht (Ed.), Swiss Academy of Natural Sciences, Bern, Orell FüssliVerlag, Zurich, 1978.8. Habicht, Walter: “Einleitung zu Band 20 der zweiten Serie” (“Introduction to vol. 20of Series II”), in [6], Series II, vol.
20, “Commentationes Mechanicae et Astronomicaead Scientiam Navalem Pertinentes”, vol. I, W. Habicht (Ed.), Swiss Academy of NaturalSciences, Bern, Orell Füssli Verlag, Zurich, 1974.9. Habicht, Walter: “Einleitung zu Band 21 der zweiten Serie” (“Introduction to vol.
20of Series II”), in [6], Series II, vol. 21, “Commentationes Mechanicae et Astronomicaead Scientiam Navalem Pertinentes”, vol. II, W. Habicht (Ed.), Swiss Academy ofNatural Sciences, Bern, Orell Füssli Verlag, Zurich, 1978.10. Burckhardt, J.J., Fellmann, E.A., Habicht, W., (Eds.): “Leonhard Euler 17071783”, Memorial Volume of the Kanton Basel City, Birkhäuser Verlag, Basel, 1983.- 27 -11. Szabó, István: “Geschichte der mechanischen Prinzipien und ihrer wichtigstenAnwendungen” (“ History of the Mechanical Principles and Their Most ImportantApplications”), Birkhaeuser Verlag, Basel-Boston-Berlin, 1977, 3rd ed., 1987.12. Mikhajlov, Gleb K.: “Leonhard Euler und die Entwicklung der theoretischenHydraulik im zweiten Viertel des 18.
Jahrhunderts” (“Leonhard Euler and theDevelopment of Theoretical Hydraulics in the Second Quarter of the 18th Century”), pp.229-241 in [7], 1983.13. Calinger, Ronald: “Leonhard Euler: The First St. Petersburg Years (1727-1741)”,Historia Mathematica, vol. 23, pp. 121-166, Academic Press, 1996.14. Calero, Julián Simón: “La génesis de la Mecánica de Fluidos (1640-1780)”,Universidad Nacional de Educación a Distancia, Madrid, 1996.15.
Darrigol, Olivier: “Worlds of Flow: A History of Hydrodynamics from theBernoullis to Prandtl”, Oxford University Press, Oxford-New York, 2005.16. Ferreiro, Larrie D.: “Ships and Science: The Birth of Naval Architecture in theScientific Revolution, 1660-1800”, The MIT Press, Cambridge, MA, 2007.17. Bernoulli, Johann: “Essai d’une nouvelle théorie de la manœuvre des vaisseaux”,Jean George König Publ., Basel, 1714. In: Opera Omnia II, reprinted: Hildesheim 1968,pp.
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