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20 MARINE REVIEW. [December 5, # M,R, . To this is to be added the whole of the connecting rod and parts moving with the crosshead to make up'm. ag, 43, a4, etc., are the angles by which the cranks 2; 3, and 4, etc., respectively lead crank 1, and # is now the angle by which crank 1 has passed. top center. : : The positions of the counterbalances, chosen to suit the design, will be known, as will also the values of the m's and a's and the 'fore and aft positions of the cranks. We may then express the moment of all the forces. (including valve gears or any other attached parts, which willbe reduced to crank radius and treated exactly as the cranks) about the for- ward counterbalance. For a balance, this moment must always be zero. Take d,,d3, 3, etc., as the distance the respective crank centers are aft of the forward counterbalance; and let n, 4, e, and r be the mass, angular advance, distance and radius of center of gravity of the.aft counterbalance. The moment then is --[R{m,4, cos f + md, cos (f + a,) + mgd, cos (9 + a4) + etc. } . -+nercos (9+ 8)]=0.. (7). For the balance required equation (7) must always be zero; and, as is well known, this will hold if it holds for two values of % not differing by: : nae a two right angles. Take 9 =0 and §= 5 Then R - §=0.nrcos 6 = -- -- (m,d, + m,d, cos a,+ m,d, cos a, + etc,)=A, (8) e Pane nr R ; ee §9=-.nrsin f= -- -- (4+ m,d, sin a, + m;d, sin a, + etc.) = B. - (9) 2 e ; As all the quantities entering into A and B are known, we can at once calculate nr and #thus, B nr= (A? + B?)%, tan B= --. (10) A In the same way we may calculate the values for the forward counter- balance. The engine will now be balanced exactly for those vertical forces which have a period equal to the time of revolution of the engine. Although Herr Schlick had read a most interesting paper in 1884, before the Institute of Naval Architects, on the "Vibrations of Steam Vessels," the real impetus to the present development of the allied subjects of vibrations of steamships and engine balancing was given by a justly famous paper read by Mr. Yarrow before the same society in 1892, on "Balancing. Marine Engines.and Vibrations of Vessels.'. There he shows how, by a calculation fundamentally the same as that in equations (7) to (10), we may determine the mass and stroke of "bob-weights," placed forward and aft of any engine; and the angular position, relative to the cranks, of the eccentrics driving them, so that the same vertical balance, as given above for the revolving counterbalances, may be attained. Ob- viously, nr and # are the same whether we adopt revolving counterbal- ances or bob-weights driven by eccentrics. The only advantage of the latter is that the excessive horizontal forces are avoided. In fact, if we counterbalance the revolving parts by revolving counterbalances, and the reciprocating parts, only, by bob-weights, an exact 'balance both of hori-, zontal and vertical forces (connecting rod infinite) thay be attained. As the throw of the eccentrics is necessarily very limita, the bob-weights become comparatively heavy; and I believe@fMr. Yarrow very soon substi- tuted the lighter revolving masses which could have a greater travel given to them, thus returning to the old counterbalancing. Instead of. driving idle bob-weights by eccentrics, to produce balancing forces by their inertia, we may evidently transform the eccentric into a crank and make the bob- weight into the piston, etc., working in a cvlinder. Its first function will not be interfered with by this change. When this is realized in a four- crank engine, we have the Y. S. T. system. I will only give the equations referring to this system. Using the same notation as before, we have for the total upward force w?R {m, cos 6+m, cos (J+ a,)+ m, cos (G+ a,)+ m, cos (g+ a,)} i) g Similar terms being added to include the mass of the valve gears, etc., reduced to crank radius. Taking the moments about the forward crank and measuring d2,d,,d, from the center of this crank, we get the equa- tion for. moments, oe w?R : /----{d,m, cos (J+ a,)+ dam, cos (0+ a,)+dym, cos (f-+a,)}. (12) To obtain a balance we must equate these to zero at two positions of a the revolution, say at @ = 0 and @ a _We then get the four equations. From (11), 9 =0.m,+m, cos a,+m, cos a,+m, cos a, =0 A ; 1: - From (11), 9 oe m, sin a,+m, sin a,tin, sin a,=0 From (12), 9=0.d,m, cos a,+d,m, cosa,;+d,m, cos a,=0 G8) 1 : ; - From ig tg d,m, sin a,+d,m, sina,+d,m, sin a, =0_} /- We may choose three of the quantities, say my, m,, eda) one '. solving the equations find corresponding values of M9):M3,-A., ag; d,'and d, will be practically determined by the design. I merely give, equations. (18) 'to show clearly the scope of the Y. S. T. system, and its limitations; and !a §o 'for a. subsequent remark on the, effect of short{period' forces, I do, Beads ntéthods of solution, as I am, not advocating. the: scheme. Those ee fS}low the matter further may consult the article on. "The: Catises of Vi Fatio of Screw Steamers," by Assistant Naval. Constructor D.. W. Taylor, U.'S.'N., in the Journal of the American Society of Naval: Engineers, Vol. III, p. 20. Constructor Taylor was the first to propose a four-crank engine with unequal crank angles in order.that the tendency t | to (11) and (12). | 2540 w2 Ro 3 Vi | ary to vibration might. be "'practically annulled." Also, see Herr. Schlick's paper, 'Further Investigations of the Vibrations of Steamers, ° Institute Naval Architects, 1894. Admirable methods of semi-graphical solution, which. can be applied to this system, including valve gears, are given by- Prof. Dalby in a paper, "The Balancing of Engines, with Special Refer- . ence to Marine Work," Institute Naval Architects, 1899, and in one by Mr. J. Macfarlane Gray. 'The Geometry of Engine Balancing," read before. the same society this spring (1901). oA : ce If the cranks are so far counterbalanced as to. make the masses of the revolving parts reduced to the crank radius--that is, the webs, pin, and part of the connecting rod--proportional to m,, mz, Mg, m4, evidently these revolving parts are balanced among themselves at all parts of the revolution. Thus the horizontal as well as the long period vertical forces ~ are balanced. I believe this horizontal balance is not usually carried out in practice, though this is the only advantage that can possibly be claimed by the Y. S. T. system 'over the older counterbalancing by revolving weights at the end of the engine. As already shown, this latter leaves unbalanced horizontal forces, but such as rarely give trouble. .The revolv- ing counterbalances have the advantage of great simplicity, and the avoidance, early in the design, of the troublesome determination of crank angles and adjustment of moving masses, with the attendant difficulty of making different parts of the crank shaft replaceable by one spare section. Also, we are not tied down to a four-crank engine. That the Y. S. T system. does. not give a perfect balance, and that the difficulties above mentioned are. not imaginary, are brought out more. strongly by the fact that the system has only been used to a limited extent, and where it was important to: claim that at least an effort had been made to solve the problem. We will incidentally examine if.a little further in treating the effects of the forces which pass through two complete cycles in one revolution of the engine. These are the forces- which principally account for the inertia at top stroke being so much greater than at bottom stroke. These short-period forces are not produced at all by revolving counter- balances, which is.much in their favor. It has long been known that the inertia from the parts reciprocating with the piston is very approximately given by Mw?R me " F=-- cop += con 20} 1 g All that has thus far been discussed neglects the very important second term of the equation just given. J may at once give the exact formula, taken from Mr. J. H. MacAlpine's paper tead before the Institute Naval Architects this summer, entitled "A Solution of.the Vibration Problem." And I may add that, as this paper gives such an excellent exposition of. the MacAlpine system, I shall quote from it very liberally in what follows. The numerical coefficients are for a connecting rod four cranks in length. 'Mw?R : ; F= (cos# + .2540 cos 29-- .0041 cos 49 + .00007 cos 69+ etc... (14), é x We must now deal with the short-period forces expressed by the terms after cos @ in the infinite series of equation (14). I think their existence and nature had not been recognized when Mr. Yarrow read his paper in 1892. As establishing their importance and, at least, usual origin in the engine, I will make a quotation from Mr. MacAlpine's paper re- ferred to above. He says: t "The second factor of equation (1)* is an infinite series, but the only term considered by Mr. Yarrow is the first, viz., cos 9. On remarking to ' Mr. Yarrow [in 1892] that the complete series should give rise to other and shorter period vibrations than those occurring at the same speed as the engine, he at once said that explained the peculiar form of wave produced by the vibrations of the boat, these waves having sharp:crests and: well- rounded hollows. Thus, from the first, we have strong evidence of the marked presence of those short-period vibrations arising from the engine --for in many of Mr. Yarrow's.experiments the propeller was diconnected. In the same year, 1892, Mr. H. C. Flood and I made observations on the, Circassia, and found not only vibrations of the same period as that of the revolution of the engine, but-also, very prominently, those of one-half and one-fourth of that period. These we called the first, second and fourth period vibrations. There were also marked indications of vibrations of the sixth and eighth period, but no trace of any odd period. This is precisely what we would expect from equation (1), which contains, besides cos 9, only the cosines of even multiples of 9. In several other respects, detailed in our paper, the observations confirmed what theory would lead us to expect; and I think they leave no doubt that these vibrations. origi- nate in the engine. Scarcely has there been a vibration diagram published - _ which has not borne the clearest evidence of those of short period; and no one who has followed the development of this whole subject could fail to be struck by the growing recognition of their importance. Indeed, in two cases which came under my own observation, and in one which. was: observed by a friend, the second period vibration. was much more promi- nent than that of the first period. I maintain that the short-périod forces must be balanced in any completely satisfactory solution of the vibration problem, and this cannot be done with the ordinary type of direct-con- nected marine engine." : Perhaps no stronger proof could be given of the importance of short- period vibrations than the fact that Herr Schlick, who has made. so many excellent observations of vibrations of ships, read an elaborate paper in 1900 before the Institute of Naval Architects, to prove that the Y. S. T.- engine could be practically balanced for second period. With this claim, , however, I cannot agree, and I believe it has not been substantiated.. The reasons for this I will give shortly. The importance of these high-period _ vibrations being now thoroughly recognized, I will give a simple method of judging quickly the degree of balance or unbalance an engine has with | regard to any particular period, and use it in two cases. This method has | been used both, by Mr. MacAlpine. and by Prof, Dalby. The second period forces and couples are. given' by expressions similar That for,the forces is + <i ae Bl arf = i af od esou! bs of id gaiiob {m', cos29-+mi*, cos2 (9+-a,)4+-m', cos2(P-+a,)+m", cos 2 OHA SHC "15) ° *Equation (14) of the present paper.