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of putting over the rudder be 'regarded as an im ulse (measured by the finite product P. dt), delivered at the stern ofp the shi-p normal to the rudder, the resistance -of the water to the rotation of the ship may be neglected, and the instantaneous centre »of the turning motion (as distinguished from the motion-ahead) is the point O on a straight line GB perpendicular to the direction of the impulse, and such that GOQGB =%an expression for the position of O of the same form as obtained before.

In this case %, =k', where k is the radius of gyration of the ship about a vertical axis through the centre of gravity, and the point O is obtained by the geometrical construction shown in fig. 64, given by Professor W. M. Rankine, where B

GL = k and is perpendicular to GB,

and the angle BLO is a right angle.

The value of I is dependent on

(I) the, distribution o weight in

the ship, being large when heavy

weights' are situated near bow and L' A .

FIG. 64.

stern, (2) the length' of the ship,

and (3) the underwater form near

the ends, being relatively large in fine ended vessels with large areas of deadwood. W is also dependent-on the shape of the shiprunderwater. ~

The handiness of a fship or her 'readiness to respond to slight. alterations in helm is mainly dependent on the relation between QXBG the moment of rudder pressure for a given angle, and I the virtual moment of inertia; lf'I isrcomparatively large, the vessel will turn slowly under helm until, gathering way, the rapidity of its angular motion becomes so large that reverse helm ma be required to limit the change of course to that desired., Unhandiness is usually experienced at low speeds (Q being then small) and also in shallow water when I is increased bg the restriction in the flow of water from one side of the ship to the other. Improvement in the handiness in these circumstances has been .obtained in certain ships with unbalanced rudders by filling in the after deadwood, the loss from the increased inertia being more than compensated b the greater turning moment due to the pressure on the after deacii wood. .

When the ship is turning steadily in a circle, 'if C (fig. 63) is' the centre of rotation, and CO.perpendicular/to the middle line of ship, the motion is equivalent to a progression ahead with speed V (which is considerably less than the initial speed), combined with a rotation about the " pivoting point ” O, which is generally situated slightly abaft the bow; the drift angle ¢> is given by the relation OG =OC tan ¢. .-The

time of turning through 180° is 1% where r is the radius OC. The forces acting upon -the ship are now-the pressure P on rudder and deadwood (if any), the centrifugal force, the thrust of the pro llers, and the pressures on the hull. The last named consist of fiiices P1 outwards before O, and P2 inwards abaft O; of these P, is usually negligible in amount; P2 cannot be directly estimated, but since work is done against it by the transverse motion of the after part of the ship, a reduction .of speed results whose amount is largely dependent on the Obliquity of motion at the centre of gravity, that- is on the drift angle ¢. Under full helm the ratio of the steady speed when, turning to the initial speed is often about 60 or 70%; but in some quickly turning ships it is less than 50 %. Of the remaining forces, the transverse com-WV2cos'¢ I . . ' .

ponent — T of the centrifugal force is known since the final diameter of turning 27 is approximately the same as the tactical diameter. To obtain P, it is to be observed that the water impinges on the rudder in a direction BF intermediate between BE (perpendicular to BC? due to the ship's motion and BD due to the form at the stern; i BF is assumed 'to bisect thefangle DBE, the effective rudder angle is approximately 0-41. The pressure on the nidder is therefore less than when helm is first put over and is further reduced on account of the diminution in the speed of the ship.

From experiments made with the object. of measuring P when turning steadily, it is found that the pressure recorded was about one-fourth of the value calculated on the assumption of the ship retaining her original speed and effective rudder angle; when helm had just been put hard over, from one-half to one-third of the theoretical Apfressure was obtained.' (See Bulletin de'l'As rociation Technique aritime, 1897; American Institution of- Naval Archs. and Mar. Eng., 1893.) The transverse forces calculated on this basis for a battleship of 15,000 tons dis lacement when turning steadily under full helm are approximately-centrifugal force 200 tons, pressure 'on' rudder 40 tons, and Q2, the transverse component of Pg, 240 tons passing through a point on the middle line about 40 ft. abaft the rentre of gravity.

The following equations applicable to the state of steady rotation can be obtained from the above considerations, neglecting P1 and the small. couple due to Rr - ' l,

A ' WV”cos1 ' I ' .

VQz=Q'i"~**§ '-ii . ., .' . (i.), ,

V ' Qz><GM=GB)<Q' . . .' . .(ii.)

From (i.) it is seen that a small tactical diameter will be obtained when Q2 is large compared with Q; from (ii.) it follows"that the point M (fig. 63) should then be near' G. These conditions are realised in a ship whose resistance to leeway is considerable but concentrated about the middle of the length, such, for example, 'as a yacht -having a deep web keel, or a boat with centre board and drop keel. In thesejnstances 'the vessel may be regarded as virtually anchored by its' keel, and the pivoting point brought to a position in closing proximity to the centre of gravity. Similarly tactical diameters. of vessels of ordinary type are reduced by diminishing the resistance to lateral motion at the after end and by increasing it amidships or forward. ' ' During the turning trials made with H.M.S. “Thunderer, ” observations were ma' e of 'the heel caused by the transverse forces brought into play whenturnin . On first uttin the 5 """ helm over 'a small inward 'heelgcaused by the preisure” Heelwhen of the rudder was observed; as the rotational' speed of tuning the ship increased this 'inclination was succeeded by 'a steady outward heel, amounting to about 1° at 7 knots s d. Theflatter is caused by the couple formed by the centrifugal fidifce and the lateral resistance diminished by= the (usually) small couple 'due"'tO'the rudder pressure. During some more recent trials carried out on the “ Yashimav ” the angle of lheel was 8%° at full speed. Similar large inclinations are generally found with modern' warships having small turning circles and high speeds and whose centres of gravity are alsosituatecl high up; at moderate speeds, however, thefheelds of small amount., On puttin the helm quickly amidships whenagturning, the opposing oouplzegue to the rudder pressure IS removed or reversed and the angle of heel momentarily increased; instances have occurred of ships with small stability and comparatively large “ rudder couples ” .capsizing through this cause. ' ' " The rudders used in ships are of two types:—(r).»Unbalanced, shown in figs. 65, 67, .68, ; and (2)balanced, shown in'figs2 66, 67 (at bow) and 69 to 74. An unbalanced rudder is in stable ecfuié librium when amidships and force has to be applied to the, .ngetf ° tiller.i§ n order toplace it at any angle to the middlaline. ri' ibm It is supported at its forward edge by means§ of'pintles'working-in U

AP . .

FIG. 65.-Cargo Vessel. FIG. 66.-Atlantic Liner. gudgeons on the stern post; and owing to its simplicity of construction. and to .its property of returning quickly to, the- middle line when the. til ler is released through any cause, .'this' ty e is preferred when the force'required' to put the rudder 'hardé over, is sufficiently moderate to enable steering to be performed' by hand or by an engine and gear of moderate size 'when steam steering is admissible. I

With high speeds and large manoeuvring powers, the unbalanced type is generally unsuitab ep; and balanced rudders are adopted AI . Ar' .

FIG. 67.¥H.M.S. “ Formidable.” FIG. 68.-HQM.S. " King' L H.M.S. " Duncan ” similar. Edward VII.”

in order to reduce the force required and the work doneito obtain lar e angles of helm. -A balanced rudder is' unstable amidships, anti, if left free comes to rest at a moderate angle on either side of the middle line. Slightly less than oneathird of the area is usually placed before the axis; in some ships in which a greater proportion has been put forward, difficulty has been experienced In bringing back the rudder to a idships. As shown in the figures, the method of support has variexdx in different ships; in many Ca'§ eS a steadying pintle has been placed at the heel, or, mid-depth, but in the latest warships the support has necessarily been taken entirely inboard. ' ' ' ' '

ln the merchant service, unbalanced rudders of the form shown in fig. 65 are generally fitted; the rudder extends'“ixp to, or above, the water-line, 'and is comparatively narrow longitudinally. Somewhat greater efficiency when usinggsmall or moderate angles of

helm is obtained with rudders of this shape; as, for a given pressure