CLASS-5A.pptx .

Department of Mechanical & Manufacturing Engineering, MIT, Manipal 1 of 25
CHAPTER 3
INERTIA FORCES
DYNAMICS OF MACHINERY

Dynamic Analysis in Reciprocating Engines-Gas Forces
Piston efforts (Fp): Net force applied on the piston , along the line of
stroke In horizontal reciprocating engines. It is also known as effective
driving force (or) net load on the gudgeon pin.
crank-pin effort: The component of Force (Fc) perpendicular to the
crank is known as crank-pin effort.
crank effort or turning movement on the crank shaft?
It is the product of the crank-pin effort and crank pin radius(r)

Crank Effort
The driving force acting on the piston is termed as piston
effort. In a vertical cylinder IC engine, following three types
of forces act:
a. Gas Force. The force due to variation of working fluid
pressure is known as gas force, or Gas force
Fg =
πD2
4
× p …….. (1)
where
D = diameter of the piston and
p = gas pressure

b. Inertia force. In an IC engine, during the first half of the
stroke, the reciprocating mass accelerates and the inertia
force tends to resist the motion. Thus the net force on the
piston is decreased.
However, during the second half of the stroke, the
reciprocating mass decelerate and inertia force opposes
this deceleration. Thus it increases the effective force on
the piston.
The inertia force of the piston is given as
………………(2)
From The velocity and
acceleration of piston
F=ma

c. Weight of the reciprocating mass.
The weight of reciprocating mass assists the piston during
its movement towards bottom dead centre (BDC).
Therefore, piston effort is increased by an amount equal to
the weight of the piston. However, when the piston moves
towards top dead centre (TDC), the piston effort is
decreased by the same amount.
Net piston effort: P = Fg + Fi  W ………….(3)

Forces acting on a slider crank mechanism
(Analytical Method)
To determine crank effort:

𝑺𝒊𝒏ø =AB/Fc
AB=Fc 𝑺𝒊𝒏ø
Fr=AB=
=Fc 𝑺𝒊𝒏ø…….(i)
cosø = 𝑨𝑪/𝑭𝒄
AC=Fc cosø
P=Fc Xcosø … … … (𝒊𝒊)
(ii)/(i)= (Fr/P)
=Fc 𝑺𝒊𝒏ø/Fc cosø
Fr=P ×tanø

Graphical Method to determine crank effort or torque
T = P x distance OY
Where OY is the distance measured between centre of
crank and a point of intersection of Y axis and extension
of connecting rod P2B

Typical questions from the chapter……

Note:
The crank effort is a function of piston effort P and crank
rotation angle. Further, the piston effort is also a function of
crank angle .
The diagram showing the crank effort or torque as a function
of crank rotation angle  for any reciprocating engine is called
crank-effort diagram or turning moment diagram.

The turning moment diagram of any engine can be plotted
if the gas pressure p is known for all positions of the crank.
The value of gas pressure can be found from a given
pressure-volume (P-V) diagram (Figure 3).
Figure 3. P-V diagram of petrol engine

Using these pressure values, gas forces can be computed
and plotted as shown in Figure 4.
Figure 4. Variation of Gas force and Inertia force

Further, the variation of inertia force due to mass of
reciprocating parts can be plotted as shown in Figure 5
with dashed line.
Figure 5. Variation of piston effort

T= piston effort x OY
Where OY is the crank effort arm length. The variation in
crank effort arm length for different crank position is
shown in figure 6.
Figure 6. Variation of crank effort arm length

Finally the turning moment diagram is shown in figure 7. A close look
at the turning moment diagram (Figure 7) shows that torque T is
entirely positive in expansion stroke of engine whereas in suction,
compression and exhaust strokes, it is negative. This indicates that in
these strokes, power is consumed. Thus there is large variation of
torque which may cause fluctuation of speed.
Figure 7. Turning moment diagram

Figure 8. Turning moment diagram for a
multi-cylinder engine.
In multi-cylinder engine, the turning
moment diagram of each cylinder is
obtained separately and they are
superimposed over each other with
starting point shifted to phase
difference of angle between respective
crank positions. A typical turning
moment diagram of multi-cylinder
engine is shown in figure 8.

Fluctuation of energy
The fluctuation of the energy is the excess energy developed by the engine
between two crank position or difference between maximum and minimum
energies is known as fluctuation of energy. TMD for a multi cylinder engine is
as shown in figure. The horizontal line AG represents mean torque line. Let
a1, a3, a5 be the areas above the mean torque line a2, a4& a6 be the areas
below the mean torque line. These areas represent some quantity of energy
which is either added or subtracted from the energy of the moving part of the
engine.
Let the energy in the fly wheel at A=E
Energy at B=E+a1
Energy at C=E+a1−a2
Energy at D=E+a1−a2+a3
Energy at E=E+a1−a2+a3−a4
Energy at F=E+a1−a2+a3−a4+a5−a6
Energy at G=E+a1−a2+a3−a4+a5−a6
Suppose greatest of these energies is at B and least at E,
Maximum energy in the fly wheel =E+a1
Minimum energy in the fly wheel =E+a1−a2+a3−a4
∴ Maximum fluctuation of energy ( E) = max. energy – min. energy
E = (E+a1 ) − (E+a1−a2+a3−a4 )
E = a2− a3+ a4

Flywheel
A flywheel is an inertial energy storage device. It absorbs
mechanical energy and serves as a reservoir, storing
energy during the period when the supply of energy is
more than the requirement and releases it during the
period when the requirement of energy is more than the
supply.
Internal combustion engines with one or two cylinders
are a typical example. Piston compressors, punch presses,
rock crushers etc. are the other systems that have
flywheel.

Types of Flywheel
Generally, three types of flywheel – disc type, web type and arm type are
most commonly used (Figure 11).

Types of Flywheel

A plot of torque vs. crank angle or turning moment
diagram of a multi cylinder engine is shown in the figure
12.
Figure 12. Turning moment diagram of a multi cylinder engine

Fluctuation of energy (Ef):
The difference between maximum and minimum kinetic energies
of flywheel is known as maximum fluctuation of energy, Ef.
A flywheel is used to control the variations in speed during each
cycle of an engine. A flywheel of suitable dimensions attached to
the crankshaft, makes the moment of inertia of rotating parts quite
large and thereby it acts as a reservoir of energy. During the
periods when the supply of energy is more than required it stores
energy, and during the period when the supply is less than required
it releases the energy.

Coefficient of Fluctuation of energy (Ke):
Ratio of maximum fluctuation of energy and work done per cycle is known as
coefficient of fluctuation of energy.
Ke = Max. fluctuation of energy / work done per cycle.
Therefore Ke =
Ef
E
Work done / cycle = Tmean x  ...from turning moment diagram (Area)
where Tmean = Mean torque and
 = angle turned by the crank in radians.
Also Work done per cycle is given: = P x 60,000/N
where P = Power in KW and N is in rpm.
P=
𝟐𝝅𝑵𝑻
𝟔𝟎𝟎𝟎𝟎
T=F×R
Work done=𝟐𝝅𝑹𝑭

Coefficient of Fluctuation of speed (Ks):
The difference between the greatest and the least angular speeds
of the flywheel is called the maximum fluctuation of speed and the
ratio of greater fluctuation of speed per cycle to the mean speed is
called Coefficient of fluctuation of speed.

Multiply and
devide by ωo

Important Assumption:

Ef =I×ωmean𝟐
×Ks
Ef = m×v𝟐×Ks

Flywheel of a Punching Press:

Flywheel of a Punching Press:
Formula’s Used:
Force for
punching

KINEMATICS OF MACHINERY
Numericals from
Chapter : Inertia Forces
End of Chapter

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