One of the first attempts at plotting the curve traversed by a ship under the action of her rudder, and the position of the ship at any instant with reference to that curve was made by Sir P. Watts in 1877 on H.M.S. "Thunderer." Similar experiments were car ried out on "Yashima" (Trans. I.N.A. 1898).
When the path of the ship is plotted, the bow of the ship is nearer to the centre of the circle or curve in which she turns, than the stern. The ship may be regarded as going ahead and turning or pivoting about a point well forward in the middle line; this is termed the "pivoting point" and the middle line is, at this point, a tangent to the curve concentric with and similar to that described by the centre of gravity of the ship. The angle between the middle line and the tangent to the curve traced out by the centre of gravity is termed the "drift angle." The maximum distance that the ship's centre of gravity travels in her original direction after the helm is put over is termed the "advance"; and the perpendicular distance between the original line of advance and the ship's position after turning through 16 points is known as the "tactical diameter." It is convenient to investigate the forces acting on the ship in three stages, (I) when the helm is put over, (2) when the ship has finally picked up her circular course, and (3) the intermediate stage when the ship is accelerating but yet has gained some of her rotational velocity. The character of the forces acting during (I) and (2) can be ascertained, and the type of motion under the complex conditions represented by (3) will consist of a gradual replacement of motion at (I) by that at (2).
As soon as the helm is put over the first effect is to alter the stream line motion at the stern. The stream lines against and in front of the rudder are expanded so that their velocity is reduced and pressure increased. This leads to a force on both rudder and deadwood ; and the eddying behind the rudder causing a reduc tion of pressure accentuates the effect. The net result is that a considerable force acts at the stern, the lateral component of which tends to turn the ship, and the component in fore and aft direc tion adds resistance to the ahead motion of the ship. The pro portion of this force due to the deadwood is unknown, but in many modern ships in which the deadwood is considerably cut away this force is small. The portion due to rudder pressure
can be calculated from the results of experiments on plates moving obliquely through the water. This force varies with the shape of the rudder and approximately as its area, the angle of helm and the square of the speed; and can be expressed by the formula, where K depends on the angle of helm and the shape of the rudder. At the maximum helm angle (usually 35°) K is about 1.2 ; but when applying this a liberal allowance should be made for the reduction of V due to resistance under helm. In actual practice the old f ormula where P is the force in lbs., A the area of rudder in sq.ft., V the speed of the ship in feet per sec. increased by a percentage to allow for propeller race and 0 the angle of helm, has been found to give good results at maximum helm angle. Both the lateral and angular movements of the ship are accompanied by the motion of a mass of water which may be regarded as virtually increasing the mass and moment of inertia of the ship.
The handiness of a ship is mainly dependent on the relation between the moment of rudder pressure for a given angle, and the virtual moment of inertia. If the inertia is comparatively large, the ship will turn slowly under helm. Unhandiness is usually experienced at low speeds when rudder pressure is small, and in shallow water when the virtual inertia is increased by the reduc tion of the flow of water from one side of the ship to the other. Improvement in such cases 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 by the greater turning moment due to the pressure on the deadwood.
When the ship is turning steadily in a circle the forces acting are the pressure on rudder and deadwood, the centrifugal force, the thrust of the propellers, and the pressures on the hull. The pres sure on the rudder is now less than when the helm is first put over, both by virtue of the fact that the effective rudder angle is less, and that the speed of the ship has decreased.