Nowacki H. Leonhard Euler and the theory of ships (794398), страница 3
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The discipline of shiptheory was included in this spectrum.Although the most important branches of ship theory had been addressed already in hisScientia Navalis, Euler found the opportunity also in his Berlin years to go deeper intocertain questions and to arrive at new results. The most important subjects during thisperiod include ship propulsion [E.116, E.413/[22]] and ship motions in oscillatorydegrees of freedom [E.415]. Several of these new studies were motivated by prizecontests of the Parisian Academy in which Euler participated with success.After some irritating quarrels with Friedrich II, Euler in 1766 returned to St.
Petersburg,where he was welcomed with open arms. During his second St. Petersburg period(1766-1783) several further studies in ship theory were performed. The most importantlasting effects probably stem from his “Théorie Complète...” [E.426/[24]], a Frenchabridged and extended translation of Scientia Navalis in popularized form. This worksoon was used as a textbook, too, in the education of French naval constructors.Euler by his lifetime oeuvre has lastingly formed and enriched ship theory. He hasbased it on the first principles of mechanics and thus placed it on a stable foundation forits future development.
He was not able to provide definitive answers to all questionsraised. Much remained open. But he had cast a basic structure of the field in whichscientific work could steadily progress.3.3.1Individual Contributions to Ship TheoryHydrostatics and Ship StabilityThe foundations of hydrostatics were laid by Archimedes in antiquity, who lived fromca.
287 to 212 B.C. In his famous treatise “On Floating Bodies” [25] he derived whatlater became known as the Principle of Archimedes, i.e., the theorem of equilibriumbetween the forces of weight and buoyancy for a floating object of arbitrary shape. Inthe same treatise, Part II, he also established a criterion of stability of this equilibriumfor a body floating at rest, though only for the special case of a body of simple shape,the axisymmetric paraboloid. His justification was that in an inclined position of thebody the couple formed by the forces of buoyancy and weight (= displacement) mustprovide a positive restoring moment for the floating position to be stable.
Else the bodyheels over further and may capsize. For further details cf. Nowacki [26]. Archimedes asfar as we know did not yet apply this criterion to actual ships.This knowledge possessed by Archimedes was almost completely forgotten for manycenturies, although fortunately a few handwritten copies of his treatise [25] in Latin andin Greek were preserved [26]. But it took many centuries, almost two millennia, beforethe essential insights in this treatise were rediscovered and applied.
It was first Stevin[27], then Pascal [28] and also Huygens [29] who resurrected hydrostatics (andaerostatics) and applied them to modern systems. Stevin, the Flemish/Dutch scientist,also first introduced the concept of hydrostatic pressure, based on the weight of thefluid column. Huygens first investigated the hydrostatic stability of simple,homogeneous bodies as they were rotated about their longitudinal axis by 360 degrees,however did not publish his results so that these appeared only posthumously in thebeginning of the 20th c. in his Collected Works [29].A few further cuts were taken at the stability problem by Hoste [30] (1698) and by LaCroix [31] (1735), but they remained still without success.
Bouguer und Euler, too, intheir prize contest treatises of 1727 did not yet offer a practically promising approach. Itwas only by means of integral calculus that they later succeeded, independently of eachother, to develop criteria for the hydrostatic stability of ships of arbitrary shape, which-7-were published in their treatises, Traité du Navire [2] and Scientia Navalis [1] in 1746and 1749, respectively. Euler’s approach will be discussed here in more detail.To understand Euler’s objectives in his Scientia Navalis, it is useful to take a look at thetitle of this work and the subtitles of the two volumes, which in free English translationare:“Ship Science or Treatise on the Construction and Operation of Ships:Vol.
I: General Theory of the Position and Motion of Bodies Floating in Water,Vol. II: Reasons and Rules for the Construction and Steering of Ships.”Euler never chose the wording “Theory of Ships”, as later authors did. But since hisopus concerns the “Science of Ships” and he aspired to furnish a general theory of theequilibrium at rest and the motion dynamics of floating bodies, applicable inshipbuilding and ship operations, as his title and subtitles claim, it is fair to state that hisobjective was the development of a Theory of the Ship, which comprises thefundamentals of mechanics for the design, construction and operation of ships. Thisclaim is justified by the systematic structure of his work, even if he was not able, beinglimited to the methods of his time, to do justice to all related questions of ship theory.He did already provide very significant contributions to the hydrostatics and stability ofships.
These subjects were treated in Chapters I-IV of Vol. I and Chapters I-III and V ofVol. II of his Scientia Navalis, both in their fundamentals (Vol. I) and their applicationsto ships (Vol. II).Euler opened Chapter I of Vol. I (Equilibrium of floating bodies) with the sentence:”The pressure which the water exerts on an immersed body at a specific point is normalto the body surface, and the force which an individual surface element experiences isequal to the weight of a cylindrical water column whose basis is equal to the surfaceelement and whose height is equal to the submergence (z) of the element under thewater surface.“This implies the defition of hydrostatic pressure p = γ z and also of the resultingbuoyancy force F by integration over the body surface SF = ∫ p dS = γ V with V = displacement volumeAs already underscored by Truesdell [3], Euler formulated here in a single sentence thenecessary and sufficient axiomatic premises on which hydrostatics are entirely based.These premises can hardly be stated more concisely and clearly.
In contrast toArchimedes, who knew only the resultant hydrostatic buoyancy force, but not thepressure, Euler (and similarly Bouguer) derived the buoyancy force as the integral ofthe hydrostatic pressure distribution. This cleared the way to calculating the buoyancyforce for arbitrary body shapes.By moment equilibrium it also holds that the buoyancy and weight resultants must actin the same vertical plane.In the course of Chapter I Euler further calculated the equilibrium floating positions ofsimple prismatic shapes (triangular, trapezoidal and rectangular cross section) through arotation of 360 degrees from the condition that the volume and weight centroids must liein the same vertical plane.
Each of these shapes has several equilibrium positions whichmay be stable, unstable or indifferent. The stability of each position requires its owninvestigation.-8-In Chapter III (Stability of the equilibrium of floating bodies) Euler dealt with thederivation of a stability criterion. He proceeded in the following steps:STEP 1: Premises and AxiomsIn his stability considerations like on the issue of equilibrium Euler proceeded from thesame premises as cited regarding the hydrostatic pressure in a fluid at rest and its actionnormal to the surface.STEP 2: Resultant Buoyancy and Weight ForcesThe resultant buoyancy force in the upright floating condition of the ship by Euler wasdetermined as the resultant of the hydrostatic pressures acting on the submerged part ofthe body surface. This force acts through the volume centroid F of the underwater hullform (in English today: Center of Buoyancy CB), which Euler called “centrummagnitudinis”.Analogously Euler combined all weight components of the ship into the resultingweight force acting through the center of gravity G.STEP 3: Volumes and CentroidsEuler defined the geometrical relationships for volumes and centroids in analytic formby integral expressions for arbitrary hull forms and floating conditions, but left theevaluation of these expressions to numerical methods.
The closed form evaluation of theintegrals succeeded only for simple, regular shapes, which he used as examples.STEP 4: Stability CriterionEuler just tersely stated:“The stability by which a floating body is retained in equilibrium will be determinedfrom the restoring moment arising when the body is displaced from equilibrium by aninfinitesimally small angle”.This basic idea stemmed from Archimedes, too, but was here applied to ships. Eulerpointed out that the ship must have a sufficient reserve of stability to be able to resist theexternal effects of heeling moments.STEP 5: Evaluation of the CriterionEuler began his derivation by considering a planar cross section of the ship (Fig.
3).The cross section was intentionally assumed to be non-symmetrical to keep thederivation general. The ship was displaced by a very small angle from the floatingposition AB to the new floating position ab. Thereby the triangle bCB was immersed,the triangle aCA emerged, so that the area centroid of the triangles moved from theemerging to the immersed side. The area centroid of the whole cross section therebymoved parallel to this shift to the immersed side (shift theorem of Archimedes). Fromthis action a restoring couple resulted, formed by the weight force (acting through G)and the buoyancy of the cross section (acting through the shifted area centroid of thecross section).
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