20 MARINE REVIEW. [November 21, objectionable only in the introduction of an unbalanced moment. The primary forces are neutralized as for the reciprocating weights, and there are no secondary forces from revolving weights. If the most complete revolving weight balance is wanted, however, the revolving weights for each crank must be separately counterbalanced. The arrangement of Fig. 20 is a decided improvement, as regards vibrations, over the customary arrangement where the cranks are at 90°. But the latter is preferable from certain practical considerations; for instance, it has no dead center, and gives a much more uniform turning moment. "A two-cylinder arrangement has recently been brought forward by MacAlpine, which is superior to that of Fig. 20 in that it eliminates sec- ondary forces. It is shown diagrammatically in Fig. 21. It gives perfect balance as regards reciprocating weights, and its unbalanced couples are athwartships instead of fore and aft, and are not likely to cause objection- able vibration. This arrangement has the practical objections mentioned above to that of Fig. 20, and a few others of its own, due to increase in number of working parts, use of rocking levers, etc. It is, however, a simple and elegant solution of the problem of complete two-cylinder_ bal- ance, and by doubling it--using two pairs of cylinders on one shaft--it can be applied to triple and quadruple expansion engines. ENGINES WITH THREE CRANKS. "For three-crank engines the force polygons are three-sided figures and the moment polygons two-sided figures. We have now reached a sufficient number of cranks to allow reduction of unbalanced forces by properly choosing reciprocating weights and crank angles. Also, in these days of triple expansion, all marine engines of importance have at least three cylinders, so the three-crank case is the first one I have discussed of practical importance. The moment polygons--two-sided figures--obvi- ously cannot be closed by variation of weight and crank angles. The force polygons are, however, three-sided figures, and as long as any two of the reciprocating weights are together greater than the third, the pri- mary force polygon can be made a closed triangle by adopting proper crank angles. The condition that two weights must be greater than the third allows a wide choice of reciprocating weights and cranks angles, and we should make a choice which will, if possible, secure balance of sec- ondary as well as primary forces and be desirable, or at least not objec- tionable, as regards the very many considerations other than vibration affecting engine design. "Now if all the reciprocating weights are made equal, and the three cranks set at 120°, we find that the primary force polygon is a closed equilateral triangle; that the secondary force polygon is also a closed equilateral triangle; that the crank angles are those fixed upon by com- mon consent as the most desirable from the many considerations other than those of balancing. "Tf it is desired to secure reciprocating weight balance as regards moments for three-crank engines, the most feasible plan is to introduce reciprocating balance weights at the ends of the engine driven by cranks or eccentrics. Such an arrangement, however, is in the four or five-crank class. Considering now revolving weights, it is obvious that if the revolv- ing weight on each crank is equal (cranks being at 120°), forces are com- pletely balanced, while there is left an unbalanced revolving moment, which is, from the nature of the case, smaller than the unbalanced recipro- cating moment. For this reason, it is not necessary as a rule to attempt further revolving balance in a three-crank engine with cranks at 120° and equal revolving weights on the cranks, which is, by the way, the usual arrangement, all three cranks and connecting rods being made the same. If complete revolving balance is aimed at, however, for moments as well as forces, the most feasible method is to counterbalance separately the revolving weight on each crank. Or the methods of revolving balance described in discussing four-crank engines may be applied. "There is a theoretically practicable arrangement of the three-crank engine, which, though unbalanced as regards both forces and moments, if properly located in the ship, is nearly balanced as regards vibration. It is of little practical value, however, owing to the fact that the nodal points of a ship cannot be accurately determined in advance of the ship's com- pletion, and in fact, are subject to changes of location with change of load- ing the ship. For three-crank engines in practice the nearest approach to balance is obtained by adopting cranks at 120°, and making the recipro- cating and revolving weights for each crank the same. A slightly better balance may be obtained by fully balancing with counter-weights the revolving weights for each crank, thus extinguishing revolving moments; but as reciprocating moments, which are usually.larger than the revolving moments, are necessarily left unbalanced, it appears hardly worth while, as a rule, to add weight, complication, and expense to a three-crank engine for the purpose of avoiding revolving moments. If a three-crank engine, balanced as regards forces, is placed about the center of the length of the ship, its unbalanced moments are least likely to set up vibrations because unbalanced moments are most harmful where a vibratory node is located, and at the center of the ship there is nearly always a loop. Torpedo boats with three-crank, equally weighted engines located about the center of length have give fairly satisfactory results as regards vibration. ENGINES WITH FOUR CRANKS. "For a four-crank engine the force polygons are four-sided figures and the moment polygons three-sided figures. Evidently, then, the balancing possibilities, so to speak, in the case of a four-crank engine, are decidedly superior to those of engines previously considered. The large and in- creasing use, of late years, of triple-expansion engines with two L. P. cylinders, and of quadruple expansion engines, has, through the use of four ctanks, increased the opportunities for balancing and the development of balanced types of engines. Progress in marine engine balance during the last ten years has been made almost entirely with four-crank engines. It is necessary, then, to consider most thoroughly the balancing possibili- ties and limitations of such engines. "Before beginning this task, I desire to call attention to the important fact that the usual spacing of cranks of 90°, found in the case of four-crank engines, is essentially inferior as regards uniformity of turning moment. If a four-crank engine has four simple, double-acting cylinders, the 90° spacing is equivalent, as regards turning moment, to two cranks at 90°. For the turning moment on a crank at 0° in such an engine is the same as that on its opposite crank at 180°. and the two cranks at 90° and 270° are equivalent to one crank with double moment at either 90° or 270°. To secure the most uniform turning moment for double-acting engines, the crank should be at or opposite the angles obtained by dividing 180", not 360, by four, the number of cranks. Thus, starting with the first crank at 0°, the second should be at 45°, or 225°; the third at 90°, or 270°: and the fourth at 135°, or 315°. This conclusion, obvious for four-crank simple engines, is readily shown to apply to quadruple engines, where about the same power is developed in each cylinder. We shall see later that the crank angles for four-crank engines, which it is necessary to adopt for balancing, approximate fairly closely to those most favorable for uniform turning, so that a balanced four-crank engine is also one with a more unifurm turning moment than if the 90° crank spacing is adopted. It may be asked why, in a three-crank engine, cranks equally spaced around the whole circumference give the best turning moment. The answer is that (since in a double-acting engine the turning moment of a crank is equiva- lent to that of a crank directly opposite) the usual three-crank arrange- ment at 0°, 120°, and 240° is equivalent to one at 0°, 120° and 60°; and in the eae the cranks are equally spaced over the first 180° of the crank circle. "The mathematical investigation of the balancing possibilities in con- nection with four-crank engines is quite feasible, though laborious. I have attacked the problem by determining first, throughout the practi- cable range of relative reciprocating weights and cylinder spacings, a large number of arrangements of crank angles, etc., for engines which are com- pletely balanced primarily, and then determined the unbalanced secondary forces and moments for these engines. It will be observed in the first place, that where the crank angles are such as to give an evidently inferior turning moment, the resultant secondary force is usually large, and the relative reciprocating weights such as would be very difficult to obtain in practice. In the second place, for arrangements where W, and W, range from 1 to 1.5, the secondary forces are not large, the crank angles approach reasonably close to those which we have seen give the most uniform turn- ing moment, and W, does not differ greatly from W4, which is assumed always as constant and equal to unity. In the third place, the secondary force varies greatly, both in direction and magnitude, while the secondary moment varies comparatively little, indicating that while we may expect, by further manipulation of weights and angles, to eliminate secondary force, it will not be possible in practice to eliminate thus the secondary moment. "The obvious inference from the results attained is that we need ex- amine in detail only arrangements where cylinder No. 4, having unit reciprocating weight and zero crank angle: Cylinder No. 3 has recipro- cating weight somewhere in the neighborhood of 1.5, and its crank angle in the second quadrant, ranging approximately from 135 to 160°. 'Cylinder No. 2 has reciprocating weight somewhere between 1 and 2, and its crank angle in the third quadrant and somewhat less than 270°. Cylinder No. 1 has reciprocating weight somewhere in the neighborhood of 1, and crank angle in the first quadrant, ranging, approximately, from 45° to 60°. "As it is not to be expected that we will be able to get rid of both sec- ondary force and secondary moment by further manipulation of weights and crank angles for four cranks, the next step in my investigation will be di- rected toward the determining of combinations where secondary forces, as well as primary forces and moments, are extinguished, and the resulting un- balanced secondary moment known. In appendix C will be found detailed the graphic methods followed which enable us to determine absolutely the crank angles and relative reciprocating weights for a four-crank engine, with any practicable cylinder spacing, for which primary and secondary forces and primary moments are extinguished. Such engines are upon the Yarrow, Schlick and Tweedy system, and have been much used abroad, it is claimed, with great success. It is seen that the cylinder axes once located, the crank angles and relative weights follow. While theoretically any order for the cylinders may be chosen, in practice, owing to the fact that the lightest weights must be on the end cranks, and that uniform turning moments are desirable, it is advisable to adopt the general arrange- ments outlined below. "For four-cylinder triple-expansion engines the two L. P. cylinders should be outside, and their moving part should be as light as possible. Each L. P. cylinder does only about half the work of the H. P. or M. P., and it is desirable to have the L. P. connecting rods, crank pins, etc., reduced in due proportion. For four-cylinder quadruple expansion en- gines the H.P. and M!P. cylinders should be outside and the M2P. and L. P. inside. Care must be taken to design the L. P. piston as light as practicable, and it is advisable to have the air pump, if worked by a lever, driven from the L. P. crosshead, as thus the L. P. reciprocating parts are virtually lightened. It is advisable to have the forward pair of cylinders as close together as practicable, and the after pair also as close together as practicable. This result is facilitated if the valves for the two outer cylin- ders H. P. and 'MP. are placed at the ends of the engine, and the valves for the two inner cylinders (M, P. and L. P. in the middle. Balance can be had without the precautions noted above, but with much greater additions of weight than would otherwise be needed. _ "With a four-crank engine, balanced as regards the reciprocating weights, there are so many ways of obtaining revolving balance that it becomes a question of selecting that one having most practical advantages as regards simplicity and cheapness. The ideal method as regards balance alone is to fit proper counter-balance weights opposite each crank. If, however, the revolving force and moment polygons are plotted, it will be found that equally complete balance can be secured with less total addition of weight. If moments are taken about the after crank as usual, a revolv- ing weight at the forward crank at a suitable angle will close the revolving moment polygon. This will, of course, mean an additional side to the revolving force polygon, but the latter can now be closed by a suitable weight at the proper angle at the after crank. In practice it is not con- venient, as a rule, to apply balance weights at the cranks, unless they are on or opposite to the crank. So the most feasible plan in practice is to fit two revolving counter-balance weights, one aft of the engine framing and the other forward. The turning wheel can usually be made use of for the attachment of one of these weights, and if the forward end of the shaft is extended two or three inches, a wheel or disk for a forward balance weight is readily attached. Once the correct force and moment polygons are plotted, the amounts and angles of these weights are readily determined, Before finally balancing revolving weights it is well to take advantage of any obvious methods for reducing the unbalanced resultants, This may ~"sen be done, for instance, by drilling holes in one or more solid crank