The determination of accurate results when making turning trials is difficult; at least it appears to be difficult to obtain identical results from sister ships or from the same ship when tried twice under apparently identical conditions. It is commonly assumed that after beginning to turn, the track of a ship for the first 180° begins as a spiral and becomes a circle, and that if no change of helm angle or engine conditions is made, the track is circular by the time 180° have been traversed, the diameter of this circular track being called the final diameter and being, of course, somewhat smaller than the tactical diameter corresponding to the first 180° of turn.
There is a very simple method of determining the final diameter of a ship when turning in a true circle, requiring no observations of a distant object or other observations of heading, and with all observations made from the ship herself. The underlying assumptions are two in number, as follows:
1. That a floating pole or buoy about which the ship turns, remains in the same place while the ship turns.
2. That the ship after a time does settle down to turn in a true circle.
Referring to Fig. 1, let BS denote the ship turning in a circle whose center is 0. Let P denote a floating pole somewhere within the circle. Let the observer at B have a plane table, pelorus, or sextant with which he can observe continuously the angle PBS, or PBL, and similarly let the observer at C be fitted to observe the angle PSB.
B and S must be in known positions, so that the length of BS and its position on the ship are known. It is convenient but not necessary to have B and S both on the center line. Extend SB to L. Let OBL= ?, OSL=y. Since 0 is the center of the circle being described by the ship the angles ? and y remain constant as long as the ship continues to move in a circle about 0, and if we can determine 13 and 7 we can complete the triangle BSO and determine the final diameter of any point of the ship.
Let BB1B2 be the circle described by B. Join OP and draw B1PB2 perpendicular to OP.
Let us call PBL, or the angle measured at the bow station, the bow angle and PSL, or the angle measured at the stern station, the stern angle. We will first consider the bow station. Evidently, PBL=OBL—PB0=?-? where ?=PBO. Now from the triangle OBP we have sin ?= sin OPB. During turning OB in a circle OB and OP remain unchanged, so that sin ? varies as sin OPB. Now the maximum value of sin OPB is unity when OPB=90°; hence sin & is a maximum when OPB is a right angle. This occurs twice during each complete circle, namely, when B is at B1 and B2.
Now when B is at B, the bow angle PB1L1= ?-?1 has a minimum value, since ?1 is a maximum value of ?, and ? is constant.
In the position B2 the bow angle PB2L2= ?-?2 has a maximum value since ?2 is again a maximum value of 0 and now is added to the constant angle (3. Then we have
PB1L1=?-?1, PB2L2=?+?2, and evidently ?1=?2.
Whence adding 2= (PB1L1+PB2L2),
?= ½ (PB1L1+PB2L2),
= (maximum value of bow angle + minimum value of bow angle.)
Evidently then the ? angle can be accurately determined if the bow observer keeps sufficient track of the bow angle to determine accurately its maximum and its minimum value during a turn. This is not difficult since the maximum and minimum values are approached and departed from slowly. Of course, if P should happen to coincide with 0 there would be no change in the bow angle. The nearer P is to the circle BB1B2 the more rapid the change from maximum to minimum and the greater the difference between maximum and minimum. It would seem desirable in practice to have P near the center rather than near the circumference.
The stern angle is treated exactly as the bow angle for the determination of y, and with ?and y known the circles described by B and S are readily drawn.
For convenience in demonstration I have taken the bow angle as measured from B forward—towards L. In practice it will nearly always be more convenient to measure the angle OBS or 180°- ?. This is particularly the case if a sextant is used.
The method described above is believed to be novel. It has been tried in practice only with an electric launch under unfavorable conditions, but fairly good results were obtained and no practical difficulties developed.
It is easy to make several circles in succession without changing helm, or engine conditions, and the successive final diameters thus obtained will give a line upon the accuracy of the method in practice. As already pointed out, with accurate observations, there are only two conditions required to make the results exact, namely, that the point P should remain at rest while the ship turns and that the ship should turn in a true circle.
Figs. 2 and 3 have been plotted to avoid the necessity of making any calculations when using this method. With BS and bow and stern angles known the value of OB can be obtained graphically from Fig. 2 in terms of BS within 2 or 3 per cent, which is quite close enough for practical purposes.
Similarly OS can be obtained from Fig. 3. The contours in Figs. 2 and 3 are nearly straight, but not quite.