The Work of the Crank Shaft
183. Besides those variations of speed which occur form stroke to stroke, which it is the business of the governor to check, there are variations within each single stroke over which the governor has of course no control. These are due to the varying rate at which work is one on the crank-shaft during its revolution. To limit them is the function of the fly-wheel, which acts by forming a reservoir of energy to be drawn upon during those parts of the revolution in which the work done on the shaft is less than the work done by the shaft, and to take up the surplus in those parts of the revolution in which the work done on the shaft is greater than the work done by it. This alternate storing and restoring of energy is accomplished by slight fluctuations of speed, whose range depends on the ratio which the alternate excess and defect of energy bears to the whole stock the fly-wheel holds in virtue of its motion. The effect of the fly-wheel may be studied by drawing a diagram of crank-effort, which shows the work done on the crank in the same way that the indicator diagram shows the work done on the piston. The same diagram serves another useful purpose in determining the twisting and bending stress in the crank.
184. the diagram of crank-effort is best drawn by representing, in rectangular co-ordinates, the relation between the moment which the connecting-rod exerts to turn the crank and the angle turned through by the crank. When the angle is expressed in circular measure, the area of the diagram is the work done on the crank.
Neglecting friction, and supposing in the first place that the engine runs so slowly that the forces required for the forces required for the acceleration of the moving masses are negligibly small, the moment of crank-effort is found by resolving the thrust P of the piston-rod into a component Q along the connecting-rod and component O normal to the surface of the guide (fig. 108). The moment of crank-effort is
Q &Mac250; CM = P &Mac250; CN = Pr sin a (1 + r cos a ) &Mac226;
where CN is drawn perpendicular to the centre line of travel of the piston, r is the crank, l the connecting rod, and a the angle ACB which the crank makes with the centre line. A graphic determination of CN is the most convenient in practice, unless the connecting rod is so long that its obliquity is negligible, when the second term in the above expression vanishes. Fig. 109 shows the diagram of cranks-effort determined in this way for an engine whose connecting-rod is 3_ times the length of its crank, and in which steam is cut off at half-stroke. The thrust P is determined from the indicator diagrams of fig. 108 by taking the excess of the forward pressure on one side of the piston over the back pressure on the other side, and multiplying this effective pressure by the area of the piston. The area of the diagram of crank-effort is the work done per revolution.
185. The friction of the piston in the cylinder and the piston-rod in the stuffing-box is easily allowed for, when it is known, by making a suitable deduction from P. Friction at the guides, at the crosshead, and at the crank-pin has the effect of making the stress at each of these places be inclined to the rubbing surfaces at an angle Ø, the angle of repose, whose tangent is the coefficient of friction. Hence O, is instead of being normal to the guide, is inclined at the angle Ø in the direction which resists the pistons motion (fig. 110); and the trust along the connecting-rod, instead of passing through the centre of each pin, is displaced far enough to make an angle Ø with the radius at the point where it meets the pins surface. To satisfy this condition let a "friction-circle" be drawn about the centre of each pin, with radius equal to a sin Ø, where a is the actual radius of the pin. Any line drawn tangent to this circle will make the angle Ø with the radius of the pin at the surface of the pin. The thrust of the connecting-rod must be tangent to both circles; it is drawn as in fig. 110, so that it resists the rotation of the pins relatively to the rod. The direction of rotation of the pins is shown by curved arrows in the figure, where the friction-circles are drawn to a greatly exaggerated scale. Finally, P (after allowing for the friction of piston-packing and stuffing-box) is resolved into O and Q, and Q&Mac250;CM, the moment of Q on the shaft, is determined. This gives a diagram of crank-effort, correct so far as friction affects it, whose area is no longer equal to that of the indicator diagram. The difference, however, does not represent the whole work lost through friction of the mechanism, since the friction of the shaft itself, and of those parts of the engine which it drives, has still to be allowed for if the frictional efficiency of the engine as a whole is in question.
186. The diagram of crank-effort is further modified when we take account of the inertia of the piston and connecting-rod. For the purpose of investigating the effects of inertia, we may assume that the crank is revolving at a sensibly uniform rate of n turns per second. Let M be the mass of the piston, piston-rod, and crosshead in pounds, and a its acceleration at any instant in feet per second per second. The force required to accelerate it is Ma/g, in pounds-weight, and this is to be deducted in estimating the effective value of P. The effect is to reduce P during the first part of the stroke and to increase it towards the end, thereby compensating to some extent for the variation which P undergoes in consequence of an early cut-off. If the connecting-rod is so long that its obliquity may be neglected the piston has simple harmonic motion, and
a = -4&Mac185;2n2rcosa= -4&Mac185;2n2x,
where x is the distance of the piston from its middle position. More generally, whatever ratio the length l of the connecting-rod bears to that of that crank r,
a = -4&Mac185;2n2r (cos a + rl2cos2a+f3sin4a)
(l 2sin2a)_/2) &Mac25
The effect is to make, on the diagram of P, a correction of the character shown in fig. 111, where the broken line cd refers to the case of an indefinitely long connecting-rod and the full line ab to the case of a connecting-rod 3_ times of the length of the crank. In a vertical engine the weight of the piston and piston-rod is to be added to or subtracted from P.
To allow for the inertia of the connecting-rod is a matter of somewhat greater difficulty. Its motion may conveniently be analyzed as consisting of translation with the velocity of the crosshead, combined with rotation about the crosshead as centre. Thence the force required for its acceleration is the resultant of three componentsF1., the force required for the linear acceleration a (which is the same as that of the piston); F2, the force required to cause angular acceleration about crosshead; and F3, the force towards the centre of rotation, which depends on the angular velocity, and is equal and opposite to the so-called centrifugal force. Let _ be the angle BAC (fig. 112), _ angular velocity of the rod about A and _ its angular acceleration, and let M be the mass of the rod. Then
and acts through the centre of gravity G, parallel to AC;
and acts at right angles to the rod through the centre of percussion H;
and acts along the rod towards A. Also,
_ = &Mac195; l 2-r2sin2a ;
- 4&Mac185;2n2rsina(l 2-r2) .
_ = (l 2-r2sin2a)_/2
The moments of these forces about C are next to be found, and to be deducted from the moment of the thrust in the connecting-rod (and, if the weight of the rod is to be considered, its moment about C is to be added) in finding the resultant moment of crank-effort.
187. If, however, the friction at the crosshead and crank-pin is to be taken account of, the whole group of forces acting on the rod must considered as follows. Compound forces equal and opposite to F1, F2, and F3 into a single force R (Fig. 113), which may be called the resultant to acceleration of the connecting-rod. If the weight of the rod is to be considered, let it also be taken as a component in reckoning R. Then rod may in any position be regarded as in equilibrium under the action of the forces Q,R, and S, where Q and S are the forces exerted in it by the crosshead and crank-pin respectively. These three forces meet in a point p in R which is to be found by trial, the condition being that in the diagram of forces, fig. 114, after the triangle POQ has been drawn, and the force R set out, the force-line S shall be parallel to a line drawn from p tangent to the friction-circle of the crank-pin, as in fig. 113. When this condition has been satisfied by trial, the value of S, which is the thrust on the crank-pin, is determined, and S. CM is the moment of crank-effort. This method is due to the late Prof. Fleeming Jenkin, who has applied it with great generality to the determination of the frictional efficiency of machinery in two important papers,1 the second of which deals in detail with the dynamics of the steam-engine. Fig. 115, taken from that paper, shows the diagram of crank-effort in a horizontal direct-acting engine,the full line with friction, and the dotted line without friction,the inertia of the piston and connecting rod being taken account of, as well as the weight of the latter. It exhibits well the influence which the inertia of the reciprocating parts has in equalizing the crank-effort in the case of an early cut-off. The cut-off is supposed to occur pretty sharply at about one-sixth of the stroke. The engine considered is of practical proportions, and makes four turns per second; and the initial steam pressure is 50 _ per square inch. It appears from the diagram that, with a slightly higher-speed, or with heavier rods, a better balance of crank-effort might be secured, especially as regards the stroke towards the crank, which comes first in the diagram; on the other hand, by unduly increasing the mass of the reciprocating pieces or their speed the inequality due to expansion would be over-corrected and a new inequality would come in.
188. When two or more cranks act on the same shaft, the joint moment of crank-effort is found by combining the diagrams for the separate cranks, in the manner illustrated by fig. 116, which refers to the case of two cranks at right angles.
Another graphic method of exhibiting the variations of moment exerted on the crank-shaft during a revolution is to draw a circular diagram of crank-effort, in which lines proportional to the moment are set off radially from a circular line which represents the zero of moment. An example of this plan is given in fig. 117, which shows the resultant moment determined by Mr. A.C. Kirk for one of his triple-expansion engines with three cranks set at 120° from each other. Curves are drawn for various speeds, giving in each case the resultant moment due to the same pressure (as determined from actual indicator diagrams) combined with the moments due to the inertia of the reciprocating parts. The line marked 0 is the steam line without inertiaor, in other words, the curve corresponding to an indefinitely slow speed. The other curves refer to the number of revolution per minute marked on them.
189. To determine the fluctuations of speed during a revolution, the resultant diagram of work done on the crank-shaft is to be compared with a similar diagram drawn to show the work done by the shaft in overcoming its own friction, and in overcoming the resistance of the mechanism which it drives. In general the resistance may be taken as constant, and the diagram of effort exerted by the crank-shaft is then a straight line, as EFGHIJKL in fig. 118. At F, F, H, I, J, and K the rate at which work is being done on and by the shaft is the same; hence at these points the fly-wheel is neither graining nor losing speed. The shaded area above FG is an excess of work done on the crank, and raises the speed of the fly-wheel from a minimum at F to a maximum at G to H the fly-wheel supplies the defect of energy shown by the shaded area below GH, by which the demand for work exceeds the supply; its speed again reaches and defects balance in each revolution if the engine is making a constant number of turns per second. In what follows it is assumed that they are only a small fraction of the whole energy held by the fly-wheel.
Let &Mac198;E be the greatest single amount of energy which the fly-wheel has to give out or absorb, as determined by measuring the wheel has to give out or absorb, as determined by measuring the shaded areas of the diagram; and let w1 and w2 be the maximum and minimum values of the wheels angular velocity, which occur at the extremes of the period during which it is storing or supplying the energy &Mac198;E. The mean angular velocity of the wheel w0 will be sensibly equal to _(w1+w2) if the range of the fly-wheel at this mean speed. Then
where I is the moment of inertia of the fly-wheel.
E = I (w12+w22)=Iw0(w1-w2)= 2E0. _(w1-w2) .
The quantity w1-w2, which we may write q, is the ratio of the
extreme range of speed to the mean speed, and measures the degree of unsteadiness which the fly-wheel leaves uncorrected. If the problem be to design a fly-wheel which will keep q down to an assigned limit, the energy of the wheel must be such that
E0 = 2q
The Moscrop recorder, alluded to in § 182, exhibits the degree of unsteadiness during a single revolution by the width of the line which it draws. On the other hand, any bending of the line implies the quite independent characteristic of unsteadiness from one revolution to another. The former is due to insufficient fly-wheel energy, the latter to imperfect governing.
190. An interesting consequence of the periodic alternations in crank-effort which occur in each revolution has been pointed out by Mr. M. Longridge.1 The fly-wheel receives its alternate acceleration and retardation through changes of the torsional stress in the shaft. If these occur at interval nearly equal to the period of tree torsional vibration which the fly-wheel possesses in virtue of the torsional elasticity of the shaft between it and the crank, strains of great amplitude will arise; and Mr. Longridge has suggested that this may account for the observed fact that engine-shaft have been ruptured when running so that the fluctuations of crank-effort occurred with one particular frequency, although the greatest effort was itself much less than the shaft would safely bear.
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