**Coefficient of Fluctuation of Speed**

The difference between the maximum and minimum speeds during a cycle is called the maximumÂ fluctuation of speed. The ratio of the maximum fluctuation of speed to the mean speed is calledÂ coefficient of fluctuation of speed.

Let N1 = Maximum speed in r.p.m. during the cycle,

N2 = Minimum speed in r.p.m. during the cycle, and

N = Mean speed in r.p.m. = (N1+N2) / 2

âˆ´ Coefficient of fluctuation of speed,

Cs = (N1-N2)/N = 2(N1-N2)/(N1+N2)

The coefficient of fluctuation of speed is a limiting factor in the design of flywheel. It variesÂ depending upon the nature of service to which the flywheel is employed. Table 1 shows the permissibleÂ values for coefficient of fluctuation of speed for some machines.

**Fluctuation of Energy**

The fluctuation of energy may be determined by the turning moment diagram for one completeÂ cycle of operation. Consider a turning moment diagram for a single cylinder double acting steamÂ engine as shown in Fig. 1. The vertical ordinate represents the turning moment and the horizontalÂ ordinate (abscissa) represents the crank angle.

A little consideration will show that the turning moment is zero when the crank angle is zero. ItÂ rises to a maximum value when crank angle reaches 90Âº and it is again zero when crank angle is 180Âº.Â This is shown by the curve abc in Fig. 1 and it represents the turning moment diagram for outstroke.Â The curve cde is the turning moment diagram for instroke and is somewhat similar to the curve abc.Â Since the work done is the product of the turning moment and the angle turned, therefore theÂ area of the turning moment diagram represents the work done per revolution. In actual practice, theÂ engine is assumed to work against the mean resisting torque, as shown by a horizontal line AF. TheÂ height of the ordinate aA represents the mean height of the turning moment diagram. Since it isÂ assumed that the work done by the turning moment per revolution is equal to the work done againstÂ the mean resisting torque, therefore the area of the rectangle aA Fe is proportional to the work doneÂ against the mean resisting torque.

We see in Fig. 1, that the mean resisting torque line AF cuts the turning moment diagram atÂ points B, C, D and E. When the crank moves from â€˜aâ€™ to â€˜pâ€™ the work done by the engine is equal toÂ the area aBp, whereas the energy required is represented by the area aABp. In other words, the engineÂ has done less work (equal to the area aAB) than the requirement. This amount of energy is taken fromÂ the flywheel and hence the speed of the flywheel decreases. Now the crank moves from p to q, theÂ work done by the engine is equal to the area pBbCq, whereas the requirement of energy is representedÂ by the area pBCq. Therefore the engine has done more work than the requirement. This excess workÂ (equal to the area BbC) is stored in the flywheel and hence the speed of the flywheel increases while

the crank moves from p to q.

Similarly when the crank moves from q to r, more work is taken from the engine than is developed.Â This loss of work is represented by the area CcD. To supply this loss, the flywheel gives up some ofÂ its energy and thus the speed decreases while the crank moves from q to r. As the crank moves fromÂ r to s, excess energy is again developed given by the area DdE and the speed again increases. As theÂ piston moves from s to e, again there is a loss of work and the speed decreases. The variations ofÂ energy above and below the mean resisting torque line are called fluctuation of energy. The areasÂ BbC, CcD, DdE etc. represent fluctuations of energy. A little consideration will show that the engine hasÂ a maximum speed either at q or at s. This is due to theÂ fact that the flywheel absorbs energy while the crankÂ moves from p to q and from r to s. On the other hand,Â the engine has a minimum speed either at p or at r. The

reason is that the flywheel gives out some of its energyÂ when the crank moves from a to p and from q to r. TheÂ difference between the maximum and the minimumÂ energies is known as maximum fluctuation of energy.

Reference A textbook of Machine Design by R.S.Khurmi and J.K.Gupta

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