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26 MARINE REVIEW. [December'5; also the moment of inertia of the whole watch; for the moment of mo- mentum of the whole watch, including the balance wheel, must remain zero. This is well known to everyone, and it holds also with the ship and engine. : = Now suppose the connecting rod infinite in length, the velocity of the pistons will be greatest at the center of their travel. Using the same sym- bols as before, the velocity at the middle of the stroke is wr, and the momentum for the moving masses of one cylinder is mwr. To simplify the explanation, suppose we are dealing with the H.P. and second I.P. which are connected by a lever. m is the same for each as the counter- balance and the parts it balances produce no torsional couple, and so may be omitted. Mvr is, then, the same for both cylinders, the centers of which stand at a distance D from one another. Take the moment of momentum about a point in the center line of the H.P. cylinder. The moment for the H.P. itself is then zero as the lever arm is zero. That for the second I.P. is mwr D. If we took the moment about a point in the plane of the cylinder center lines, at a distance a from that of the H.P., we would have: For the P.H. moment = mur Xa For the second I.P. moment = mwr X (D -- a) Total moment = mwr xX D. Thus the value of the moment does not change with the position of the point around which we take it. If @ is the distance of any small mass M of the ship from the axis around which it revolves torsionally, and w, is the greatest angle by which it departs from its undisturbed position, the linear displacement of M from its mean position has a maximum value pw, and the velocity with which it passes through its: mean position is ~v,w, since the vibration is of the same period as the engine revolution, The greates momentum is, then, Mp¢,w, and the moment of momentum Mpv,w xX p = Mp?¢,w. For the whole ship this moment of momentum becomes ~,w2Mp? = y,wl where I is the moment of inertia of the ship about the longitudinal axis for these vibra- tions If the total moment of momentum of engine and ship is to be zero they must oppose one another, and we must have v1 = mwrD Oe, oe mrD a Dp, =. oF e . (20) The linear amplitude of this vibration will be greater the further we are from the axis. ,At the side of the ship it will be mrDb Yb= ao (21) where b is the half breadth of the ship. Mr. MacAlpine says that the time of roll of the ship and the action of the water will have no effect on this vibration. While he is right in the first part, I think he is mistaken as to the action of the water. If the sections of the ship were parts of circles in the immersed portions, the ship might vibrate torsionally with almost no disturbance of the water. But in a very full ship a considerable movement of the water would result, and still more if bilge keels were fitted. This action would be roughly equivalent to an increase of the moment of inertia of the ship and thus would diminish, I believe, often to a very considerable extent, this vibra- tion. Now, from (d), p. 25, we get m for-H.P. and 2d I.P. cylinders ££6,802 Ibs. = 3.04 tons. m for Ist I.P. and L.P. cylinders = 7,556 lbs. = 3.38 tons. The velocity of the-H.P. piston (connecting rod infinite) when the forward crank is % past top stroke is oe wr sin 9 =e and the moment of momentum for the parts attached to this crank, 3.04 wrD sin 9. : Similarly, since the after crank leads by 90°, its moment of momentum at the same instant is 3.88 wrD cos 9. The total moment of momentum for both cranks is: wrD (3.04 sin § + 3.38 cos 9) == (3.04? + 3.382)% X wrD sin (9 + a). Hence in using equations (20) and (21), instead of m we may put (3.042 + 3,382)%, In twin screw ships this effect will at times be doubled, and we will use 2 (8.042 + 3.382)% = 9.08 tons. _ Mr. MacAlpine calculates the rigid body vibration which twin engines, Figs. 7 to 18, would give the U. S. S. Alabama. Her engines are of about the same power. Displacement load (draught) = 11,562 tons, __ Radius of gyration he guesses as = 30 ft., and it cannot be greatly different. Then I = 11562 & 302. b = about 36 feet = 36 & 12 in: m = 9.08 tons, from above. ro it. D-= 6 ft. (actually 5 ft, 10 in.) Then from equations (20) and (21) 9.08 X 2x6 "11,662 Xx 302 1b = 1.05 x 10-5 x 36 X 12 .0045 inch. Total movement at side of ship = .0045 & 2 -- .009°in./or less than 1/100 of an inch. a To force home the smallness of this movement he calculates the ap- parent movement\of the sight of a gun on a target three miles distant. This evidently is-- : 1 3 xX 5280 ---- Ka 44 inches. 100 36 = 1.05 x 105 He next shows that the effects of the second and higher period forces in producing rigid body vibrations will be exceedingly small, even in com- parison to that of the very small first-period vibration just calculated. Will these effects be increased or diminished as we change the size of the ship? Let us compare two. cases where every dimension of ship A is twice that of:ship B, machinery as well as ship; boiler pressure and piston speed to remain the same. Displacement 22... 2s Ca yikes Ship B=% of that of Ship A. Radius of gyration .........-. Jol ee = yy z L G=Disp: << radius of eyrat.2) 3.6 sce see S vs 4 f, ccank taditis«,:.... i ee ee i a yy z D; cyl, centers athwattship.......... 0... yy e m, moving masses (linear dimensions?)... i : Wetted suttice -40 5.2. S \y a LYCP, Gs cylinder areas)... .c-. 2 oe 3g . Grate and, heating surface. 7... .2.2.-.5. 3: s 14 Hence 7, for ship B %ymx%rxX%D mrD gel I Thus #, is not altered, and the I.H.P. per square foot of wetted surface also remains the same for both ships. Possibly the small ship would have its scantlings rather lighter than in exact proportion to the dimenions of the two ships as supposed, in which case #, for ship B would be a trifle larger than ?, for ship A. The total linear movement, 2",b, would therefore be little more than half in ship B what it is in ship A. In very high-powered small warships the I.H.P. per square foot of wetted surface might be four, five, or six times as great as in large warships, but an increase such as this would not bring up 2¥,b, to more than a very minute value, epecially as in the very high-powered boat the engine would be pressed to its greatest limit of piston speed, and the scantlings kept as light as possible--that is, the power would be taken out of as small and as light an engine as prac- ticable, which both go to reduce #,. Also the larger #, is, the more pow- erful will be the action of the water in opposing the vibration. Thus, even in this, the worst case, the solution is still quite satisfactory. : MacAlpine finally refers to twin screws with unbalanced engines as forming a much worse case, in which there is usually no noticeable effect and never one of serious magnitude. In support of this statement of his, and as strongly supporting his whole contention, I may add another quota- tion from Mr. Yarrow's celebrated 1892 paper: "As a further proof that the vibration is due to the machinery, I may mention that two years ago I made a passage to the United States in one of the very fast twin-screw steamers. I selected a berth in the central portion of the vessel, thinking it a good position for comfort, but the vibra- tion was found to be so excessive that after five days it was scarcely bear- able to the passengers whose berths, like my own, were situated at the points of greatest vibration. The vibration was found to vary periodically. When the two low-pressure pistons were descending at the same time it was excessive; but when one low-pressure piston was ascending and the other descending it was entirely neutralized." Here we have the whole matter brought to a crucial test. Mr. Yar- row's berth being at the center of the vessel was not only at the point of maximum vertical, but also of maximum torsional vibration. No doubt the berth was at the outside of the vessel where the movement from a given angle of torsion would-be greatest. The engines were exceedingly far from being balanced, for when acting together they gave rise to vertical vibrations "scarcely bearable." When the vertical forces were exactly counteracting one another, realizing precisely the effect of MacAlpine's proposed engine, but with a greatly exaggerated leverage between the opposing forces, the vibration was "'entirely neutralized." The evidence could not be stronger or freer from doubt; and it enhances its value a thousand fold that it was given by an engineer of the highest ability who had been specially studying and experimenting on this very subject for years previously to 1890. It there had been a sensible torsional vibration it would no doubt have been noted. The conditions were entirely favor- able to their being set up. The engines would usually. run so nearly at the same speed that elastic torsional vibrations would have ample time to reach their maximum development before the favorable phase passed. The rigid body rotation requires no time to grow; it appears at once of such amplitude as will keep the moment of momentum always zero. Yet none was noted though the unbalanced forces and the lever arm between the engines was so large. In what immediately precedes, it has been stated that with the ordi- nary engine elastic torsional vibrations do not occur. It may be objected that Herr Schlick, in his 1895 paper, before the Institution of Naval Architects,-speaks of elastic torsional vibrations, giving formulas and dia- grams. He does not mention the length of ship on which his observations were made, but judging from the engine revolutions (150) one is let to conclude it was not a very large ship. His formula, which is equivalent to equation (17), would probably have given not less than 1,500 vibra- tions per minute. He omits to give a numerical example. But his dia- grams of torsional (?) vibrations are marked 150 per minute. Mr. Mal- lock, in the discussion, calls his attention to the great discrepancy between the calculated and observed periods. Herr Schlick's reply is entirely unsatisfactory. He says the difference is due to the small torsional rigidity of the steel hull. "Every ship builder," he says, "knows that the rigidity ofa ship's hull against torsion is relatively small."' To reduce torsional vibrations from 1,500 to 150 per minute would require a reduction of (i001? 4 torsional rigidity in the proportion |----| =---~! (1500 J 100 As the ship was a German war vessel, it would have a steel deck, and, consequently, a torsional rigidity more nearly 80 or 90 per cent. of the corresponding steel tube. I have little doubt that what Herr Schlick observed was the 'first period rigid body rotation spoken of above. I regret very much that in this discussion I have been compelled to point out what I consider errors-in Herr-Schlick's papers, atid I do not wish in any way to detract from the great credit which is due him for the original work he hes done in the study of hull vibration and engine bal- ancing. I am sure that if he were present he would appreciate that these criticisms are really a tribute to the prominent position which he occupies in connection with this subject, and that they simply represent an honest = 7, for ship A.