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18 MARINE REVIEW. CURVE OF STABILITY. BRIEF METHOD OF DETERMINING IT--PRACTICE OF SCOTCH SHIP BUILDERS : IN THIS IMPORTANT PROBLEM OF SHIP DESIGN. By George Crouse Cook, A. M. J. N. A. The importance and value of investigating the stability of a proposed ship design, and so modifying the work as to give such stability as has been proved satisfactory in actual practice, has been generally acknowl- edged by modern designers. But in many instances, and especially in the smaller. offices not equipped with the mechanical integrator, calcula- tions beyond the initial stability or metacentric height are wholly over- looked, and the actual stability at large angles of inclination is assumed to be that of some similar ship. This is obviously unreliable and often leads to the production of a vessel of such stability as to be not only a source of annoyance to the crew and officers, but also a constant danger. Defective stability may possibly be corrected by the introduction of some form of ballast, but this involves the expense of transporting an unnecessary dead weight and the corresponding reduction in carrying capacity, which lessens the earning power of the ship. This subject is fully understood by the economical, practical Scotch ship builders along the Clyde, and the following comparatively short, convenient method of plotting the stability curve (a curve of righting arms for a constant displacement and varying angles of inclination) of a design is in use among them for such general merchant work as does not require elaborate and exhaustive calculations. Its chief value lies in its convenience for estimating the stability of a preliminary design, which is to be again, worked over accurately; but when the design is for a ship similar to some form for which the corresponding curve is known, and with which the general character of the new curve may be judiciously checked and compared, it may be taken as giving the true stability curve, fs | 1 «ol 1 GIy Ll (a) : oO 20 40 z » 160 Bo 100 : 120 : xs xX oe OMe ese Ae qeT ese fe -------- + _ ANGLES OF INCLINATION. FIGURE 1, na : : ; da --- This method is based on certain propositions of the geometry of 'stability and is composed of the results of investigations of three con- ditions of the ship: First, when the ship is upright; second, when heeled to an angle of 90°; third, when at some convenient intermediate condition, as the angle corresponding to the point of submersion of the deck edge, or about 40° to 50°. These investigations are so performed as to involve the calculations for displacement, center of buoyancy and the metacenter only, and therefore the method may be used with fair 'confidence by anyone familiar with the conventional displacement sheet, while in the hands of an experienced calculator it can be brought to give absolute results. : FIRST CONDITION, THE SHIP UPRIGHT, _ Assume the displacement sheet to be completed and from it take the metacentric height, GM, which is applied in the following manner as a component of the stability curve: On a line OX, Fig. 1, set off from O to some suitable scale, as %4-in.=1°, the degrees of inclination of A Ww. | i rex FIGURE 2G. ood Fe the ship from the upright position which is represented by O. Then at the point G, corresponding to 57.3° on O X, erect a perpendicular, and on this set up the metacentric height, GM, to the scale chosen to repre- sent the righting arms, say 3-in.--1 ft. Connect O and M by a straight line O M; this will then determine the initial direction of the curve, as it is a tangent to the curve of stability at O. S [June 6, SECOND CONDITION, THE SHIP HEEL TO AN ANGLE OF 90° In Fig. 2 let WL be the horizontal water line, W' L' the water line when upright and A K the original vertical center line. Let V_represent the volume of the constant displacement W' AL', or WTL'L, and H be the volume of the total water excluding body. H is found by ex- tending the operations of the displacement sheet for the upright condi- tion to include the whole body, and Q, the center of this volume above the keel, is found in this work by the same method as B, the center ot buoyancy of the upright displacement W' A L'. Now a series of water lines are laid off from the lowest horizontal tangent at T, the displace- ments calculated, and a displacement curve drawn to include the region about WL; on this curve the value of B is plotted and the correspond- ing ordinate gives the height of W L above the tangent, which deter- mines the thickness of the volume AWLK. The height of -B', the center of buoyancy of WTL'L, must be determined, and then the horizontal position of the center of volume of the layer AW LK is found by independent calculation to be, say, in the vertical through C. Now if Z, the point of intersection of a vertical through B' and the original vertical center line, which is now a horizontal through G, be determined, the righting arm at 90° may be measured. If V and H/2 were equal, WL would coincide with A K, and B' would be in the vertical through Q; but-V is here less than H/2 by the volume of the layer AW LK, which is equal to (H/2--V), therefore B' is out from the vertical through Q, a distance which is dependent on the volume and position of the center of the layer. Therefore by moments: QZxXV=0 C (H/2--V). © Z=0 € (A/2 V--!1). Then, QZ+0Q G=GZ. (The negative righting arm at 90°). Plotting Z gives the horizontal position of B' as it is in the same vertical; this then completely determines the position of the center. Now the inertia of the non-symmetrical water plane, W L, about-a fore- and-aft line through Z, is calculated, and by substitution in the formula for the height of M.above B 1B M=I/V) the specific value of B' M' is found. On plotting this value of B' M' in Fig, 2 it is seen that Z M' represents the vertical distance between G and M', the metacentric height. This righting arm and metacentric height are now used as addi- tional components of the desired curve. At the point on O X, Fig. 1, corresponding to- 90°, GZ is set off below the line, as it is the arm of an upsetting couple; then at 57.3° on from 90° or at 147:3° the value of Z M' is set down on'a perpendicular from O X; connect M' and 90°; then, as in the first condition, the direc- tion or slope of the curve at 90° is determined by this line. But:the curve is known to pass through the point Z,-and therefore if*a line is drawn through Z parallel to 90-M', the curve must be tangent to it at this point Z. ; THIRD CONDITION, THE SHIP INOLINED TO THE "DECK EDGE" ANGLE, The righting arm for this condition could be found by the well- known Barne's method of polar integration, but the following substitu- tion, while involving the same principles, may be used with but a little loss in accuracy and a great saving in time. This loss, however, is not of great importance, as those leading features of the curve, the initial slope and the range, are so well defined by the conditions of the tan- gents. ' On a complete body plan, as Fig. 3, a water line, WL, is drawn through the lowest point of the deck edge so that it cuts off, as estimated by the eye, a displacement equal to that of the upright. This can be done with fair accuracy by carefully comparing the portion of each section which is cut off, as WS W', with that part added, L WL'; the means of these as functions of the volume must be equal, as the dis- placement taken from the left hand side must be added on the right so that the total volume remains the same. For the usual ship-shape form, where the deck edge is not submerged at an inclination of about 40°, W L will fall below the point O, a distance, measured on the vertical, varying from 6 in. to 12 in. Now the horizontal position of the center of buoyancy for the form of displacement WAL must be found. This is most simply done by the principle of moments, where the volume of the transferred displace- ment LSL' or W'SW into the distance its center of gravity has moved constitutes the moment, and this moment divided by the total volume of the displacement. gives the distance the: center of buoyancy has shifted. For the purpose of finding: the moment of this volume: of wedge-shaped displacement the shifted portion of each' section, as L'S L, is assumed to be a triangle; this is quite justifiable in many cases, and where the line of the section is much curved within the limits of the