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Taking n = 12,000,000 lbs. per square inch and neglecting the effect of the inertia of the shaft, equation (19) gives :

Period of vibration = 0.021333 second.

Equation (1), Appendix II., shows the method of taking into account the inertia of the shaft.

In this case :

Moment of inertia of the shaft = Period of vibration = 0.021339 second.

14 lb.-inch units and the

In this example the masses are small, but nevertheless the inertia of the shaft has very little effect on the period of vibration. This clearly shows that the shaft may be neglected in calculations concerning high-speed engine sets. In fact, even in the case of a marine-engine the effect of inertia of the whole length of shaft on the period of vibration is very small and may be neglected.

Example 2.--In order to obtain satisfactory running in a highspeed engine, and also to reduce to a minimum the vibrations set

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up in the foundations and in the surrounding ground, the enginebuilder strives to obtain a good turning-moment on the crankshaft, and also, as nearly as possible, a balanced engine. In many cases it is not feasible to have more than three cranks on the engine; he then naturally places them at 120° to one another, makes the reciprocating parts in each line the same weight, and also brings the cylinders as close together as possible. The result of this last arrangement is that the crank-shaft usually turns out to be stiff as far as torsional vibration is concerned.

- Taking the most common case, in which there is first in order on the shaft the three-throw engine, then the fly-wheel, and then the dynamo-armature, all the reciprocating masses of the engine may be considered to act in the same plane as those on the crank next to the fly-wheel. Further, in the calculation of the periods of vibration of the system, for all practical purposes the lengths of shaft may be taken as shown in Fig. 13.

In taking the length b, the crank-web C and part of the crank-pin are neglected, but in all ordinary cases the error involved by doing

so would be very small, particularly as the effect of the reciprocating parts in most cases is itself but small. The frequency and period of vibration of a system such as this can be calculated by means of equations (29) to (33).

As a numerical example the following particulars may be taken :

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By means of equation (15) Appendix II., the moment of inertia of the reciprocating parts is found to be 12.3 lb.-foot units, and the following values, for substitution in equation (32), are arrived at:

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Substituting these values in equation (31), k = 223 or 1,123; therefore frequency of vibration 355 or 179, and period of vibration 0.0282 or 0.0056 second.

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Neglecting the reciprocating parts and finding the frequency for the system consisting only of the fly-wheel and the armature and the shaft between them, equations (23) and (24) give :—

Frequency = 35.5

Period = 0·0282 second,

a result which might have been anticipated from the small value of the moment of inertia of the reciprocating parts compared to that of the fly-wheel and armature. In practice, therefore, it is unnecessary to take account of the reciprocating parts. The practical problem is thus much simplified.

For safety, the period of the engine in this case should be about 0.043 second; therefore the speed should be not more than about

450 revolutions per minute for a single-acting engine or 225 revolutions per minute for a double-acting engine. If there were another dynamo on the same shaft, the reciprocating parts would be neglected and the frequency-equations would be worked out similarly for the three masses-the two armatures and the flywheel-but very often in a case such as this the fly-wheel and the first armature are so close that in practice they may be considered as rigidly connected; this reduces the system to one containing two masses.

Example 3.-The following is an actual case, in which trouble was caused through vibrational effects.

The plant was arranged as shown diagrammatically in Fig. 14. The engine was a three-crank, double-acting triple-expansion set, with cylinders side by side and running at 360 revolutions per

Figs. 14.

ENGINE.

FLYWHEEL.

N°I. DYNAMO.

No 2.DYNAMO.

minute. The work done in the high-pressure, intermediate, and low-pressure cylinders respectively was 1, 1.7, and 1 when running non-condensing and 1, 1, and 1.7 when running condensing. The engine drove two dynamos and there was a fly-wheel between the first dynamo and the engine. As this set was first designed there was a distinct flicker of the lights from dynamo No. 2, but very little flicker of those from dynamo No. 1. That this was not due to an electrical fault was shown by putting the armature of dynamo No. 1 into dynamo No. 2, and that of No. 2 into No. 1, when the result remained unaltered, i.e., the lights of dynamo No. 2 flickered while those of No. 1 were comparatively steady. The flicker was found to synchronize with the revolutions of the engine, and the effect was more noticeable when the engine was working non-condensing than when working condensing. The armature of dynamo No. 1 was practically rigidly connected to the fly-wheel; consequently these two could be considered as one mass. It was found that the period of vibration of the outer armature and the combined mass of the inner armature and the fly-wheel was 0.08 second. In this particular engine, both when working non-condensing and when working condensing, one crank did more work than the other cranks, in the proportion of 1.7 to 1.0. As a result, this crank

set up disturbing vibratory effects if there was approximate synchronism between its beats and the period of vibration of the system. The period of the beats of this crank-the engine being double-acting-was 0.084 second, or almost equal to the period of vibration of the armatures and fly-wheel. Evidently the flicker of the lights was due to this synchronism, and since the combined mass of the fly-wheel and inner armature was considerably greater than that of the outer armature, the changes of velocity of the latter were greater than those of the inner one, thus making the effect on the lights more noticeable in the case of the outer machine. It was further noticed that, taking the portion of the shaft between the intermediate crank-pin and the fly-wheel, there was another vibratory system whose period was 0.16 second, or al octave below the period previously referred to. This would reinforce the effect when the engine was working non-condensing -the intermediate cylinder giving the greatest effort.

other hand, with the engine working condensing, it was the lowpressure cylinder which gave the greatest effort, and since the period of the vibratory system, consisting of the part of the shaft between the low-pressure crank-pin and the fly-wheel, did not synchronize in any way with the beats of the engine, the effect was not reinforced, as was the case when the engine ran non-condensing. The difficulty might have been overcome by equalizing the efforts on the three cranks, for then the period of the beats of the engine would have been second. The method adopted, however, was to increase the diameter of the dynamo-shaft sufficiently to bring the period of natural vibration of the system out of approximate synchronism with the beats of the one crank which was giving the greatest effort, and the result was entirely satisfactory.

Example 4.-In considering the case of a single-crank engine with a fly-wheel on each side, such as is often seen in gas-engine work, the system may be taken as consisting simply of the two wheels and the equivalent length of shaft between them. It has been previously pointed out that when there are two loads on a freefree shaft there is a node at that point of the shaft which corresponds to the centre of gravity of the two inertias. Consequently in this arrangement there will be a node at, or very near, the centre of the crank; hence it follows that the reciprocating masses will have little or no effect. This system would then be dealt with by means of equations (19) to (28).

The Paper is accompanied by a sheet of drawings, from which the Figures in the text have been prepared; and by the following Appendixes.

APPENDIXES.

APPENDIX I.

LONGITUDINAL VIBRATIONS.

The following are some cases not considered in the text. The same notation is used as in the portion dealing with this class of vibrations.

If in the case of a free-free shaft carrying two loads (p. 380) the mass m of the shaft, although small, be not altogether negligible, the following equation is obtained place of equation (11):

k =

: ( E, 0 )*(M, +M2+3m)+{(M, + ¿m) (M2+¿m) (M, +M2+¿m)} * .

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In this equation it is assumed that I denotes the total length of the shaft and that the loads are at the ends.

Single Load on a Shaft fixed at both ends (Fig. 15).—Supposing the shaft to be composite, as in Fig. 15, consisting of two portions (E,, σ1, l1) and (E2, σ2, 11⁄2), the load M being at their junction, the frequency-equation is—

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2 T

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and are the frequencies of the fixed-free vibrations of M on the parts

2 π

(E1, σ1, 11) and (E2, σ2, 11⁄2) respectively. Equation (2) may be written

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If the shaft be uniform in section and material throughout, so that

(4)

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Obviously L is the length of the fixed-free shaft which has the same frequency

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