Notice Board


Thursday, March 17, 2016


In practice, there are two following types of cases where reciprocating engine mechanism is used :

  1.  An internal combustion engine or a steam engine which is used as a prime mover to drive         generators, centrifugal pumps, etc.
  2. A punching machine which is driven by a prime mover like electric motor.

In both these cases either a variable torque is supplied where demand is a constant torque or demand is variable torque whereas constant torque is supplied. In both these cases there is mismatch between the supply and demand. This results in speed variation. In case of generators, speed variation results in change in frequency and variation in voltage. On the other hand, punching machine requires energy at small interval only when punching is done. To supply such large energy at the time of punching, motor of high power shall be required. At the same time, there will be large variation in speed. To smoothen these variations in torque, flywheel is used which works as a energy storage. This results in usage of low power motor in punching machine.
After studying this unit, you should be able to
  1.  explain the method of drawing turning moment diagram for a prime mover,
  2.  determine the fluctuation of energy in a cycle,
  3.  determine the power of prime power, and
  4. determine mass moment of inertia of a flywheel and design it.

Turning Moment Diagram
If the effect of correction couple is ignored, the approximate turning moment
M = (Gas force + Inertia force) O2 D
The diagram which is plotted for ‘M’ against crank angle ‘’ is called turning moment
diagram. This diagram can be plotted progressively as explained below :
(a) There are two forces, i.e. gas force and inertia force.
Gas force = p *Piston area
where p is the gas pressure.
The variation in the gas force will be due to the change in pressure. The gas
force and inertia force have been plotted in Figure 4.4(a) for all the four
(b) The net force is the resultant of gas force and inertia force.

The turning moment in the suction stroke and exhaust stroke is very small. In case of compression stroke and expansion stroke turning moment is higher. In compression stroke, energy is to be supplied and in expansion stroke, large amount of energy is available. By surveying the turning moment diagram, it is observed that the energy is supplied in three strokes and energy is available only in one stroke. Therefore, three strokes, i.e. suction stroke, compression, and exhaust stroke the engine is starving of energy and in expansion stroke it is harvesting energy. At the same time it is observed that there is large variation of turning moment during the cycle. The variation in the turning moment results in corresponding variation in speed of the crank.
Turning Moment Diagram of a Multicylinder 4-stroke IC Engine In case of multi cylinder engine there will be more expansion strokes. For example, in the case of three cylinder engine, there will be three expansion strokes in each cycle. In case of 4 cylinder 4-strokes engine there will be four expansion strokes. Therefore, in multi cylinder engine there will be lesser variation in turning moment as compared to single cylinder engine and consequently there is expected to be less variation in speed. The turning moment diagram for a multi cylinder engine is expected to be as shown in Figure. Therefore the variation in the turning moment reduces with the increase in the number of cylinders.

It has been discussed in the preceding section that fluctuation of energy results in fluctuation of the crank shaft speed which then results in fluctuation of the kinetic energy of the rotating parts. But the maximum permissible fluctuation in speed of the crank shaft is determined by the purpose for which the engine is to be used. Therefore, to keep the maximum fluctuation of speed within a specific limit for a given maximum fluctuation of energy, a flywheel is mounted on the crank shaft.

Mass Moment of Inertia of Flywheel for an IC Engine The function of the flywheel is to store excess energy during period of harvestation and it supplies energy during period of starvation. Thereby, it reduces fluctuation in the speed within the cycle. Let w1 be the maximum angular speed and w2 be the minimum
angular speed.
Let I be the mass moment of inertia of the flywheel.
Neglecting mass moment of inertia of the other rotating parts which is negligible in
comparison to mass moment of inertia of the flywheel.
Maximum kinetic energy of flywheel

Let V1 be the maximum tangential velocity at the radius of gyration and V2 be the minimum tangential velocity at the radius of gyration.
                                                         V1=kw1  and  V2=kw2
It can be observed that
(a) The flywheel will be heavy and of large size if ΔE is large. The value of ks is limited by the practical considerations. Therefore, single cylinder 4-stroke engine shall require larger flywheel as compared to the multi-cylinder engine.
(b) For slow speed engine also the flywheel required is larger in size because of high value of I required.
(c) For high speed engines, the size of flywheel shall be considerably smaller because of lower value of I required.
(d) If system can tolerate considerably higher speed fluctuations, the size of flywheel will also be smaller for same value of ΔE.



In three-dimensional homogeneous coordinate representation, when a point P is translated to P' with coordinated (x,y,z) and(x',y',z') can be represented in matrix form as:



Unlike 2D, rotation in 3D is carried out around any line. The most simple rotations could be around coordinate axis. As in 2D positive rotations produce counter-clockwise rotations.
Rotation in term of general equation is expressed as
R = Rotation Matrix
Rotation matrix when an object is rotated about X axis can be expressed as:

Rotation matrix when an object is rotated about Y axis can be expressed as:
Rotation matrix when an object is rotated about Z axis can be expressed as:

Scaling an object in three-dimensional is similar to scaling an object in two-dimensional. Similar to 2D scaling an object tends to change its size and repositions the object relative to the coordinate origin. If the transformation parameter are unequal it leads to deformation of the object by changing its dimensions. The perform uniform scaling the scaling factors should be kept equal
NOTE: A special case of scaling can be represented as reflection.
if the value of SxSy or Sz be replaced by -1 it will return the reflection of the object against the standard plane whose normal would be either x axis, y axis or z axis respectively.

In 3D-reflection the reflection takes place about a plane whereas 2D reflection it used take place about an axis. The matrix in case of pure reflections, along basic planes, viz. X-Y plane, Y-Z plane and Z-X plane are given below:

Transformation matrix for a reflection through X-Y plane is:

Transformation matrix for a reflection through Y-Z plane is:

Transformation matrix for a reflection through Z-X plane is:

Rotation in three dimension is more complex than the rotation in two dimensions. Three dimensional rotations require the prescription of an angle of rotation and an axis of rotation. The canonical rotations are defined when one of the positive x,y,zcoordinate axis is chosen as the axis of rotation. then the construction of rotation transformation proceeds just like that of a rotation in two dimensions about the origin.

Steps to be performed
  1. Translate origin to A1
  2. Align vector with axis (say, z
    1. Rotate to bring vector in yz plane 
    2. Rotate to bring vector along z-axis 
  3. Rotate line P1Pabout z-axis which is already aligned with the Rotation axis.
  4. Reverse steps 2
  5. Reverse step 1


Some orientations of a three dimensional object cannot be obtained using pure rotations; they require reflections. In three dimensions, reflection occur through a plane. By analogy with the previous discussion of of two dimensional reflection three dimensional reflection through a plane is equivalent to rotation about an axis in three dimensional space out in to four dimensional space and back into three dimensional space. For pure reflection the determinant of the reflection matrix is identically -1
Steps to be performed
  1. Translate origin to A1
  2. Align vector with axis (say, z
    1. Rotate to bring vector in yz plane
    2. Rotate to bring vector along z-axis 
  3. Reflect the line P1P through the standard y-z plane.
  4. Reverse steps 2
  5. Reverse step 1


The simplest of the Parallel projections is the orthographic projection, commonly used for engineering drawings. They accurately show the correct or true size and shape of single plane face of an object. orthographic projections are projections onto one of the coordinate planes x=0, y=0, z=0.The matrix for projection onto the z plane is
Notice that the third column (the z column) is all zeros. Consequently, the effect of the transformation is to set the zcoordinate of a position vector to zero.
Similarly, the matrices for projection on to x=0 and y=0 planes are