Turning Moment Diagram For Multi Cylinder Engine Pdf

Chapter 16 : Turning Moment Diagrams and Flywheel 565

16.1.16.1.16.1.16.1.16.1. IntrIntrIntrIntrIntroductionoductionoductionoductionoductionThe turning moment diagram (also known as crank-

effort diagram) is the graphical representation of the turningmoment or crank-effort for various positions of the crank. It isplotted on cartesian co-ordinates, in which the turning momentis taken as the ordinate and crank angle as abscissa.

16.2.16.2.16.2.16.2.16.2. TTTTTurururururning Moment Diagram fning Moment Diagram fning Moment Diagram fning Moment Diagram fning Moment Diagram for a Singleor a Singleor a Singleor a Singleor a SingleCylinder Double Acting Steam EngineCylinder Double Acting Steam EngineCylinder Double Acting Steam EngineCylinder Double Acting Steam EngineCylinder Double Acting Steam EngineA turning moment diagram for a single cylinder

double acting steam engine is shown in Fig. 16.1. The verticalordinate represents the turning moment and the horizontalordinate represents the crank angle.

We have discussed in Chapter 15 (Art. 15.10.) thatthe turning moment on the crankshaft,

P 2 2

sin 2sin

2 sinT F r

n

= + 565

TTTTTurururururningningningningningMomentMomentMomentMomentMoment

DiagramsDiagramsDiagramsDiagramsDiagramsand Flywheeland Flywheeland Flywheeland Flywheeland Flywheel

16FFFFFeaeaeaeaeaturturturturtureseseseses1. Introduction.

2. Turning Moment Diagramfor a Single CylinderDouble Acting SteamEngine.

3. Turning Moment Diagramfor a Four Stroke CycleInternal CombustionEngine.

4. Turning Moment Diagramfor a Multicylinder Engine.

5. Fluctuation of Energy.

6. Determination of MaximumFluctuation of Energy.

7. Coefficient of Fluctuationof Energy.

8. Flywheel.

9. Coefficient of Fluctuation ofSpeed.

10. Energy Stored in aFlywheel.

11. Dimensions of the FlywheelRim.

12. Flywheel in Punching Press.
566 Theory of Machines

Fig. 16.1. Turning moment diagram for a single cylinder, double acting steam engine.

where FP = Piston effort,

r = Radius of crank,

n = Ratio of the connecting rod length and radius of crank, and

= Angle turned by the crank from inner dead centre.From the above expression, we see

that the turning moment (T ) is zero, when thecrank angle () is zero. It is maximum whenthe crank angle is 90 and it is again zero whencrank angle is 180.

This is shown by the curve abc inFig. 16.1 and it represents the turningmoment diagram for outstroke. The curvecde is the turning moment diagram forinstroke and is somewhat similar to thecurve abc.

Since the work done is the productof the turning moment and the angle turned,therefore the area of the turning momentdiagram represents the work done perrevolution. In actual practice, the engine isassumed to work against the mean resistingtorque, as shown by a horizontal line AF.The height of the ordinate a A represents themean height of the turning moment diagram.Since it is assumed that the work done bythe turning moment per revolution is equalto the work done against the mean resistingtorque, therefore the area of the rectangleaAFe is proportional to the work done againstthe mean resisting torque.

Notes: 1. When the turning moment is positive (i.e. when the engine torque is more than the mean resistingtorque) as shown between points B and C (or D and E) in Fig. 16.1, the crankshaft accelerates and the workis done by the steam.

For flywheel, have a look at your tailors manualsewing machine.
Chapter 16 : Turning Moment Diagrams and Flywheel 567 2. When the turning moment is negative (i.e. when the engine torque is less than the mean resisting

torque) as shown between points C and D in Fig. 16.1, the crankshaft retards and the work is done on thesteam.

3. If T = Torque on the crankshaft at any instant, and

Tmean = Mean resisting torque.

Then accelerating torque on the rotating parts of the engine

= T Tmean4. If (T Tmean) is positive, the flywheel accelerates and if (T Tmean) is negative, then the flywheel retards.

16.3. Turning Moment Diagram for a Four Stroke Cycle InternalCombustion EngineA turning moment diagram for a four stroke cycle internal combustion engine is shown in

Fig. 16.2. We know that in a four stroke cycle internal combustion engine, there is one workingstroke after the crank has turned through two revolutions, i.e. 720 (or 4 radians).

Fig. 16.2. Turning moment diagram for a four stroke cycle internal combustion engine.

Since the pressure inside the engine cylinder is less than the atmospheric pressure duringthe suction stroke, therefore a negative loop is formed as shown in Fig. 16.2. During the compressionstroke, the work is done on the gases, therefore a higher negative loop is obtained. During theexpansion or working stroke, the fuel burns and the gases expand, therefore a large positive loop isobtained. In this stroke, the work is done by the gases. During exhaust stroke, the work is done onthe gases, therefore a negative loop is formed. It may be noted that the effect of the inertia forces onthe piston is taken into account in Fig. 16.2.

16.4. Turning Moment Diagram for a Multi-cylinder EngineA separate turning moment diagram for a compound steam engine having three cylinders

and the resultant turning moment diagram is shown in Fig. 16.3. The resultant turning momentdiagram is the sum of the turning moment diagrams for the three cylinders. It may be noted that thefirst cylinder is the high pressure cylinder, second cylinder is the intermediate cylinder and the thirdcylinder is the low pressure cylinder. The cranks, in case of three cylinders, are usually placed at120 to each other.
568 Theory of Machines

Fig. 16.3. Turning moment diagram for a multi-cylinder engine.

16.5. Fluctuation of EnergyThe fluctuation of energy may be determined by the turning moment diagram for one complete

cycle of operation. Consider the turning moment diagram for a single cylinder double acting steamengine as shown in Fig. 16.1. We see that the mean resisting torque line AF cuts the turning momentdiagram at points B, C, D and E. When the crank moves from a to p, the work done by the engine isequal to the area aBp, whereas the energy required is represented by the area aABp. In other words,the engine has done less work (equal to the area a AB) than the requirement. This amount of energyis taken from the flywheel and hence the speed of the flywheel decreases. Now the crank movesfrom p to q, the work done by the engine is equal to the area pBbCq, whereas the requirement ofenergy is represented by the area pBCq. Therefore, the engine has done more work than therequirement. This excess work (equal to the area BbC) is stored in the flywheel and hence the speedof 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 isdeveloped. This loss of work is represented by the area C c D. To supply this loss, the flywheel givesup some of its energy and thus the speed decreases while the crank moves from q to r. As the crankmoves from r to s, excess energy is again developed given by the area D d E and the speed againincreases. As the piston moves from s to e, again there is a loss of work and the speed decreases. Thevariations of energy above and below the mean resisting torque line are called fluctuations ofenergy. The areas BbC, CcD, DdE, etc. represent fluctuations of energy.

A little consideration will show that the engine has a maximum speed either at q or at s. Thisis due to the fact that the flywheel absorbs energy while the crank moves from p to q and from r to s.On the other hand, the engine has a minimum speed either at p or at r. The reason is that the flywheelgives out some of its energy when the crank moves from a to p and q to r. The difference between themaximum and the minimum energies is known as maximum fluctuation of energy.

16.6. Determination of Maximum Fluctuation of EnergyA turning moment diagram for a multi-cylinder engine is shown by a wavy curve in Fig.

16.4. The horizontal line AG represents the mean torque line. Let a1, a3, a5 be the areas above themean torque line and a2, a4 and a6 be the areas below the mean torque line. These areas representsome quantity of energy which is either added or subtracted from the energy of the moving parts ofthe engine.
Chapter 16 : Turning Moment Diagrams and Flywheel 569Let the energy in the flywheel at A = E,

then from Fig. 16.4, we have

Energy at B = E + a1Energy at C = E + a1 a2Energy at D = E + a1 a2 + a3Energy at E = E + a1 a2 + a3 a4Energy at F = E + a1 a2 + a3 a4 + a5Energy at G = E + a1 a2 + a3 a4 + a5 a6 = Energy at A (i.e. cycle repeats after G)

Let us now suppose that the greatest ofthese energies is at B and least at E. Therefore,

Maximum energy in flywheel

= E + a1Minimum energy in the flywheel

= E + a1 a2 + a3 a4 Maximum fluctuation of energy,

E = Maximum energy Minimum energy= (E + a1) (E + a1 a2 + a3 a4) = a2 a3 + a4

Fig. 16.4. Determination of maximum fluctuation of energy.

16.7. Coefficient of Fluctuation of EnergyIt may be defined as the ratio of the maximum fluctuation of energy to the work done

per cycle. Mathematically, coefficient of fluctuation of energy,

EMaximum fluctuation of energy

Work done per cycleC =

The work done per cycle (in N-m or joules) may be obtained by using the following tworelations :

1. Work done per cycle = Tmean where Tmean = Mean t