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Historical Roots of the Theory of Hydrostatic Stability of Ships

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Historical Roots of the Theory of Hydrostatic Stability of Ships 8th International Conference on the Stability of Ships and Ocean Vehicles Escuela Técnica Superior de Ingenieros Navales 1 HISTORICAL ROOTS OF THE THEORY OF HYDROSTATIC STABILITY OF SHIPS Horst Nowacki, Technical University of Berlin (Germany) Larrie D. Ferreiro, Imperial College London and Office of Naval Research London (U.K. / U.S.A.) Abstract The physical principles of hydrostatic stability for floating systems were first pronounced by ARCHIMEDES in antiquity, although his demonstration examples were limited to simple geometrical shapes. The assessment of stability properties of a ship of arbitrary shape at the design stage became practically feasible only about two millennia later after the advent of infinitesimal calculus and analysis. The modern theory of hydrostatic stability of ships was founded independently and almost simultaneously by Pierre BOUGUER (“Traité du Navire”, 1746) and Leonhard EULER (“Scientia Navalis”, 1749). They established initial hydrostatic stability criteria, BOUGUER’s well-known metacenter and EULER’s restoring moment for small angles of heel, and defined practical procedures for evaluating these criteria. Both dealt also with other aspects of stability theory. This paper will describe and reappraise the concepts and ideas leading to these historical landmarks, compare the approaches and discuss the earliest efforts leading to the practical acceptance of stability analysis in ship design and shipbuilding. 1. INTRODUCTION (“Scientia Navalis”, 1749, [3], [4]) were the founders of modern ship stability theory, who Human awareness of the significance of ship quite independently and almost simultaneously stability for the safety of ocean voyages is arrived at their landmark results for hydrostatic probably as ancient as seafaring. An intuitive, stability criteria. BOUGUER developed the qualitative understanding of stability and of the theory and introduced the terminology of the risks of insufficient stability must have existed metacenter and the metacentric curve. EULER for millennia. The foundations for a scientific defined the criterion of the initial restoring physical explanation and for a quantitative moment, which for hydrostatic stability assessment of hydrostatic stability were laid by amounts to an equivalent concept. The full ARCHIMEDES in antiquity (“On Floating implementation of computational methods for Bodies”, [1]). Yet despite many important evaluating these criteria and their acceptance contributions and partially successful attempts by practitioners took even several decades by scientists in the early modern era like longer.Paradoxically this chain of events raises STEVIN, HUYGENS, and HOSTE among two nearly contradictory questions, which this others it took until almost the mid-eighteenth paper will address: century before a mature scientific theory of ship hydrostatic stability was formulated and o Why did it take so long for these published. Pierre BOUGUER (“Traité du formalized, quantitative criteria for the stability Navire”, 1746, [2]) and Leonhard EULER of ships of arbitrary shape to be pronounced? 8th International Conference on the Stability of Ships and Ocean Vehicles Escuela Técnica Superior de Ingenieros Navales 2 o When at last the discoveries were made, EULER in terms of the initial restoring why then did two independent, but equivalent moment. BOUGUER went beyond this in solutions suddenly appear almost at the same several practical aspects. The intriguing time? question remains: How did two scientific minds work independently to come to rather The knowledge required to evaluate the equivalent results? What were their sources, stability of a ship rests on many concepts and their background, their approach, their logic, requires many autonomous discoveries to be their justification and verification? Which were made and to be brought into concerted their unique original thoughts and how did they application. These include: differ? o The idea of conceptual experiments in To answer these questions it is not sufficient to dealing with mechanical systems. look only at their final results and conclusions, o The abstraction of thinking in terms of but it is necessary to examine more closely the resulting forces and moments of weight and methodical approaches taken to the ship buoyancy (“lumped effects”) substituted for stability issue and to compare the trains of distributed effects of gravity and pressure. thought by which the individual authors arrived o The axiom of force equilibrium, here at their results. In this article we will review the between weight and buoyancy force (Principle developments that led to this historical stage of ARCHIMEDES). when modern hydrostatic stability theory was o The axiom of moment equilibrium in a founded. system at rest. o A definition and a test for system stabi- lity. 2. PRECURSORS o A method for finding the combined weight and center of gravity (CG) for several 2.1 Archimedes weight components. o A principle for finding the resultant ARCHIMEDES of Syracuse (ca. 287-212 buoyancy force and its line of action (through B.C.), the eminent mathematician, mechanicist the center of buoyancy, CB). and engineering scientist in antiquity, is also o A method for calculating volumes and the founder of ship hydrostatics and hydrostatic their centroids, first for simple solids, then for stability, which he established as scientific arbitrary ship shapes. subjects on an axiomatic basis. ARCHIMEDES o An analytical formulation of a stability was brought up in the early Hellenistic era in criterion (for infinitesimal and for finite angles the tradition of Greek philosophy, logical rigor of inclination). and fundamental geometric thought. ARCHIMEDES was well educated in these Although the physical principles of hydrostatic subjects in Syracuse and very probably also stability were already established by spent an extended study period in Alexandria, ARCHIMEDES, it took a long time before the the evolving Hellenistic center of science, at analytical formulation of stability criteria could the Mouseion (founded in 286 B.C.). There he be pronounced for the general case of ships, met many leading contemporary scientists, e.g., essentially by means of calculus, and before DOSITHEOS, ARISTARCHOS and ERATO- numerical evaluations became feasible. STHENES, with whom he maintained lifelong BOUGUER and EULER were the first to find friendships and scientific correspondence. an analytical criterion for initial stability, Steeped in this tradition of Greek geometry and BOUGUER in terms of the metacenter. mechanics, ARCHIMEDES learned how to 8th International Conference on the Stability of Ships and Ocean Vehicles Escuela Técnica Superior de Ingenieros Navales 3 excel in the art of deductive proofs from first premises usually based on conceptual models, ARCHIMEDES preceded his work on i.e., models of thought, by which physical hydrostatics by a number of other treatises reality was idealized. But ARCHIMEDES was establishing certain axioms of mechanics: also unique, as many legends on his engi- neering accomplishments tell us, in applying The Law of the Lever: practical observation to test his scientific In his treatise “The Equilibrium of Planes” hypotheses and to develop engineering appli- ARCHIMEDES first deals with the equilibrium ations, although these achievements are not of moments about a fulcrum in a lever system. mentioned in his own written work. Although he claims to deduce this principle from geometric reasoning alone, it is actually We are fortunate that many, though not all, of understood today that the law of the lever is ARCHIMEDES’ treatises are preserved, all equivalent to the axiom of moment equilibrium derived from a few handwritten copies made in in mechanics. Second, he introduces the the Byzantine Empire during the 9th and 10th concept of “centroids” of quantities (areas, centuries and transmitted to posterity on volumes, weights) into which the quantities can circuitous routes to resurface essentially be “lumped” as concentrated effects so that through the ARCHIMEDES revival during the moment equilibrium is retained. Third, he Renaissance. Luckily, one Greek manuscript, proposes a method for finding the “compound which was later lost, had been translated into centroid” of a system of components, e.g., a Latin by the Dominican monk Willem van center of gravity. Finally, he proves the critical MOERBEKE. This translation, which was “centroid shift theorem” i.e., a rule for the shift published in 1269 and was later named Codex of the system centroid when some quantity is B, also contained ARCHIMEDES’ treatise “On added to, removed from or shifted within the Floating Bodies”, Books I and II. It became the system. All of these concepts and results are only accessible reference to ARCHIMEDES’ essential physical principles as prerequisites for work on hydrostatics for many centuries until his work on hydrostatics. in 1906 most surprisingly an old 10th century palimpsest was rediscovered in a Greek Quadrature: monastery in Constantinople by J. HEIBERG In his treatise “The Quadrature of the Parabola” [5]. This palimpsest under the writing of a 12th ARCHIMEDES illustrates by the example of century monk of a Greek prayerbook had the parabola how the Greeks determined areas retained significant traces of the rinsed off text and volumes of elementary shapes without the of a Greek manuscript from ARCHIMEDES availability of calculus. Here he uses a method, including the Greek version of “On Floating well-known since EUDOXUS [7], based on an Bodies”. ARCHIMEDES’
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