Nowacki H. Leonhard Euler and the theory of ships (794398), страница 8
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It was only after the introduction of steam propulsionthat more advanced inventions were able to achieve technical and commercial success.Euler’s physical explanations and calculations for such propulsion systems most likelywere not known to those inventors. It was only much later that his thoughts werereassessed and were apt to be taken into consideration in modern theories of shippropulsion. As successful inventors of some of the earliest, patented and more maturesolutions deserve to be named:-3.5Steam ship with paddle wheel: Symington (1802), Robert Fulton (1807)Screw propeller: Ressel (1826), Ericsson (1834-36), Smith (1838)Jet propulsion: Ruthven (1851), Seydell (1856) and the cannon boatWATERWITCH (1867)ManeuveringIn ship theory the term „maneuvering“ generally comprises all motions of the shiptaking place in the horizontal plane as translations (parallel and normal to the ship’slongitudinal axis) and as rotation about the vertical axis (yawing).
Let the coupling withother degrees of freedom of the ship be neglected here as a first approximation. Thusmaneuvering does include the dynamic behavior of the ship on a straight course, also inoblique inflow, and in turning maneuvers. Great technical and practical significance wasattributed above all to the optimization of sailing performance of the big sailing vesselsof the 18th c. in different wind conditions and directions.
This is why studies of shiptheory already in the early 18th c. had been concerned with maneuvering theories(Renau [19], Huygens [20], Johann Bernoulli [17]). Euler continued the work of theseprecursors by his own contributions, above all on the dynamics and aero- andhydrodynamics of sailing ships, and gave essential new momentum to this topic area.The most essential results can again be found in the “Scientia Navalis” [1] and in the“Théorie Complète” [24]. The synopses by Habicht in [7], [8], [9] are very helpful forthe understanding, too.The analysis of ship maneuvers requires physical and analytical knowledge in thefollowing areas:- 22 --The magnitude and direction of forces and moments acting on hull, sail and rudder,also for wind directions obliquely to the course.The dynamics of the system, especially under the influence of inertia, sailing rigand resistance forces.The solution of the equations of motion.Euler concerned himself thoroughly with all these aspects.
His determination of forcesand moments here again suffered from the weaknesses of Newton’s impact theory, buthis contributions to the system dynamics and the integration of the equations of motionremained unaffected by this. They were in part breakthroughs and are valid until today.These applications of mechanics and infinitesimal calculus to ship motions inmaneuvers belong to the first practically and technically successful contributions bymodern dynamics and fluid mechanics.In Scientia Navalis [1], vol. I, Chapter II Euler developed an approach for the rotationalmotion of an extended body system of arbitrary mass distribution and about a given,fixed axis of rotation through the body center of gravity. Here he still presumed “freerotation” about the axis through the CG, i.e., he disregarded any “bearing reactions”that may result from the coupling with other inertia effects stemming from simultaneousrotation about other axes (by deviational moments).
These assumptions would holdstrictly only if the principal axes of inertia were chosen as coordinate system. For shipshe chose an orthogonal coordinate system through the center of gravity with thelongitudinal axis being horizontal in the ship center plane. Thereby the simplification inneglecting the couplings holds in good approximation. Then by integration over allmass elements of the ship he obtained the angular momentum M in a turning motion(where r = distance of the mass element dm from the center of rotation):M= Θ (dω/dt),where Θ = ∫ r2 dm= the axial mass moment of inertia,and ω = angular velocity.The mass moment of inertia was first introduced by Euler in this place and for thepresent purpose (he named it „momentum inertiae“). With these preparations the motionof the maneuvering ship could now be derived by integration of the translational androtational laws of dynamics, if the external forces, i.e., thrust and resistance, and theirmoments, were known.
However Euler still neglected the influence of thehydrodynamic mass moment of inertia which in a turning maneuver may be ofcomparable magnitude as the mass moment of inertia of the body.Euler now first calculated a few examples for the axial mass moments of inertia ofsimple, homogeneous bodies.
Then he investigated the motions of the ship on a straightcourse before the wind, e.g., in a stopping maneuver with the ship slowed down by itsresistance with all sails reefed, or in accelerating the ship from rest by a given sail force.Finally he considered also the case of oblique wind, i.e., with the wind acting obliquelyto the course and the rudder laid for coursekeeping at steady speed. In this applicationthe drift angle was estimated empirically from plausible assumptions and was assumedto be speed independent.
The heel angle under wind load was estimated hydrostaticallyand the sail force was adjusted thereto. In systematic series investigations the sail forceswere then investigated for wind directions before the wind, with quartering winds andpointing high. Even the deflection of the sail cloth was taken into account, thougheverything only according to impact theory. With these “polar curves” of the rig (intoday’s terminology) Euler was also able to deal with nautical problems, e.g., findingfavorable strategies for cruising against the wind.Regarding the placement of masts and the arrangement of sail area, a subject to whichEuler returned several times, he applied the dynamics of maneuvering to arrive at verypractical recommendations. The masts and their sails should be placed in such a waythat the sail force resultant through the sail area centroid would act slightly abaft the- 23 -center of action of the transverse hull resistance (“lateral plan centroid”) so that thecouple of these two forces would turn the forebody into the wind.
This tendency couldbe compensated by minor rudder action to keep the ship on course. By contrast shipstending to drift leeward suffered from increasing drift angles, resistance increases anddifficulties in coursekeeping. Euler was able to explain such observed phenomenaplausibly by his mechanics of maneuvering. The prediction of forces by magnitude anddirection was more difficult. The centers of action of the resulting forces on the sails inair and on the hull under water were estimated in accordance with impact theory to lie inthe centroids of the respective areas, thus in the sail area centroid and the lateral planarea centroid, respectively.The simplifications made in the choice of axes of inertia continued to concern Euler forsome more time. It was only later in the context of his analysis of the arbitrary rotationalmotion of a body and in connection with the equations of motion of the gyroscope [56],[57] that he arrived at a general solution for the arbitrary rotation of a body about itscentroid, formulated in terms of the “principal axes of inertia” through the center ofgravity.
If the rotational motion of the body was represented with reference to thesethree orthogonal axes, then the deviational moments of inertia would vanish and theequations of motion with uncoupled inertia terms would hold exactly in this referenceframe. The principal axes of inertia were defined and determined by Euler according toan idea by Segner (1707-1777), who had published this in 1758.The theory of ships in this field owes much gratitude to Euler’s efforts and insights thathave remained of classical, lasting value throughout the field of mechanics, i.e., wellbeyond the initially motivating field of ship motions.3.6Ship motionsThe ship, considered as a rigid body, has three oscillatory degrees of freedom in whichinertial and restoring forces or moments exist so that a periodic oscillation may arise:Heaving (translation parallel to the vertical axis), rolling (rotation about the longitudinalaxis) and pitching (rotation about the transverse axis of the ship).
These motions aredesignated in this subsection as “oscillatory ship motions”, or briefly as “ship motions”.Since in all of these degrees of freedom hydrostatic forces or moments are involved, it israther easy to date the time since when the treatment of these oscillations becamefeasible, viz., only after Bouguer and Euler had created a foundation for the calculationof such restoring forces or moments acting on the ship in its position displaced fromequilibrium.
Thus it is no coincidence that both, again independently and almostsimultaneously, published first theories on oscillatory ship motions, viz., in theirmonumental principal works Théorie du Navire [2] und Scientia Navalis [1] (appearedin 1746 and 1749). The solutions and even calculation methods proposed by both arenot equal, but in practice equivalent. Thus we may limit ourselves here to the narrationof Euler’s contributions.Fortunately in addition a very substantial part of the correspondence between Euler andJohann as well as Daniel Bernoulli has been conserved (“Commercium Epistolicum”[58], contained in Series IVa of [6]), which contains several letters dating from 1738 to1740 on the subject of ship oscillations.
Euler, who at this time was writing a chapter onship motions in Scientia Navalis, had succeeded in convincing his teacher Johann andhis friend Daniel Bernoulli to work on similar tasks. Therefrom resulted, in addition toEuler’s own results, also some publications by the Bernoullis during the same period,e.g., by Daniel Bernoulli [59] in the years 1738/39.The studies initially concentrated on the determination of natural frequencies andperiods of ship oscillations in order to predict or avoid resonances.
For this purpose itwas required to know inertia and restoring forces or moments, which could now bepredicted quite realistically by available methods. (However the influence of- 24 -hydrodynamic masses, which may be of considerable magnitude in many degrees offreedom, was not yet taken into account).All three scientist - and Bouguer likewise- had noticed the analogy between a physicalpendulum, which had been investigated earlier by Galileo and Huygens, as it occurs in apendulum clock, and the ship moving in an oscillatory degree of freedom.
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