Mechanics of Machines Lab

MECHANICS OF
MACHINES (OEL)
SUBMITTED TO:
Sir Rashid Sajid
SUBMITTED BY:
2019-ME-102, 2019-ME-104
2019-ME-108, 2019-ME-123
DEPARTMENT OF MECHANICAL ENGINEERING

UNIVERSITY OF ENGINEERING AND TECHNOLOGY,
LAHORE
MOM-L OEL
To balance two rotating disturbing masses with two
balancing masses rotating with same radii in different
planes.
APPARATUS:
 Balancing Apparatus
 Weights
 Nuts
THEORY:
Balancing:
Balancing is defined as the correction or removal of unwanted
inertial forces or unbalanced forces. In a balancing process,
unbalanced forces are either eliminated out completely or reduced
to a minimum level. It is an attempt to improve a body’s mass
distribution.
Why do we Balance components?

Unbalanced forces or inertial moments can cause a body to
undergo reversed/repeated stress. Thus, an unbalanced body can
fail in fatigue before its expected life cycle. Even though
reversed/repeated stresses are not created, unbalancing increases
the stress in the components, thus making a compromise on their
safety and life cycle. Thus, it can noise, wear and tear.
Main objective of Balancing:

The main objective of undergoing a balancing process is that the
system’s centre of gravity remains stationery during its complete
revolution. In some cases, balancing of parts is achieved couples
involved in the acceleration of the different moving parts.
Methods of Correcting Unbalance:

One common method of correcting unbalance is by removal of
material from the heavier part of the component using the
technique of drilling or milling. If not, the alternate method is to add
suitable mass at the required place to nullify the effect of
unbalanced forces.
Static & Dynamic Unbalance:

Static Balancing:
 Static balancing can be defined as the balance of forces due to
action of gravity.
 When the axis of rotation is in line with the body’s axis of rotation,
the body is in static equilibrium.
Dynamic Balancing:
 balance achieved due to the action of inertia forces is called
dynamic balance.
 When the resultant moments or couples, which are involved in
the acceleration of different moving parts is equal to zero, the
body is dynamically stable.
 Meeting the dynamic equilibrium conditions automatically
meets the conditions of static equilibrium.
Balancing of Rotating Masses:

It is a process of balancing a rotating body by applying another
mass(masses) at a suitable position to nullify the effect of
unbalanced centrifugal or inertial force(forces) caused by the other
mass/masses.
PROCEDURE:
 First check the stability of the mechanism by turning it ON.
 Then the known masses of random values were chosen from the
wight box and added to the 2nd and 4th wheel as the disturbing
masses.
 Now when the apparatus is turned ON the apparatus shows
visible vertical vibrations due to unbalanced masses.
 Apply the equations derived below to determine the
magnitudes and orientations of the balancing masses.
 Mount the balancing masses on 1st and 3rd wheel.
 Check the apparatus for static as well as dynamic balance by
turning it ON.
DERIVATION:
The configuration of the planes containing the balancing masses is shown above
in the diagram. In this case, we have four different parallel planes containing 2
balancing and disturbing masses at alternate positions. We need to derive an
equation for calculating the magnitude and angle of 2 balancing masses when
all other values are known. So, we consider here as:
M1 = Disturbing mass 1
M2 = Disturbing mass 2
B1 = Balancing mass 1
B2 = Balancing mass 2
Mc1 = It stands for cos component of disturbing mass 1
Ms1 = It stands for sin component of disturbing mass 1
Similarly, cos and sin components of other masses follow the same nomenclature.
θd1 = Angle of Disturbing mass 1
θd2 = Angle of Disturbing mass 2
θ1 = Angle of Balancing mass 1
θ2 = Angle of Balancing mass 2
Now, we will use the conditions of equilibrium to determine the magnitudes and
respective angles of the balancing masses B1 and B2. The derivation is shown
below.
Using ΣFx = 0
Mc1 + Mc2 + Bc1 + Bc2 = 0
Bc1 + Bc2 = −(Mc1 + Mc2 ) − − − (1)
Similarly, using the force balance along y axis:

ΣFy = 0
Ms1 + Ms2 + Bs1 + Bs2 = 0

Bs1 + Bs2 = −(Ms1 + Ms2 ) − − − (2)
Now, summing all the moments about x-axis to 0, we get the equation:
ΣMx = 0
Bc1 (lB1 ) + Bc2 (lB2 ) + Mc2 (lM2 ) = 0
Bc1 (lB1 ) + Bc2 (lB2 ) = −Mc2 (lM2 ) − − − (3)
Similarly, ΣMy = 0
Bs1 (lB1 ) + Bs2 (lB2 ) + Ms2 (lM2 ) = 0

Bs1 (lB1 ) + Bs2 (lB2 ) = −Ms2 (lM2 ) − − − (4)
From Equation (1):

Bc1 = −(Mc1 + Mc2 ) − Bc2 − − − (5)
Putting value of Equation (5) in (3), we get:

[−Mc1 − Mc2 − Bc2 ](lB1 ) + Bc2 (lB2 ) = −Mc2 (lM2 )
On further simplifying above equation, we get:
Bc2 (lB2 − lB1 ) = [Mc1 + Mc2 ](lB1 )−Mc2 (lM2 )
[𝐌𝐜𝟏 + 𝐌𝐜𝟐 ](𝐥𝐁𝟏 )−𝐌𝐜𝟐 (𝐥𝐌𝟐 )

𝐁𝐜𝟐 =
(𝐥𝐁𝟐 − 𝐥𝐁𝟏 )
Using Equation (1), we have:

𝐁𝐜𝟏 = −(𝐌𝐜𝟏 + 𝐌𝐜𝟐 +𝐁𝐜𝟐 )
Similarly, we solve for Bs2 , by using equation (2) and (4). We get:
[𝐌𝐬𝟏 + 𝐌𝐬𝟐 ](𝐥𝐁𝟏 )−𝐌𝐬𝟐 (𝐥𝐌𝟐 )

𝐁𝐬𝟐 =
(𝐥𝐁𝟐 − 𝐥𝐁𝟏 )
𝐁𝐬𝟏 = −(𝐌𝐬𝟏 + 𝐌𝐬𝟐 +𝐁𝐬𝟐 )
Now, for finding B1 and B2, we take the resultant of the sin and cosine
components obtained from above equation. So, the resultant will be obtained
as:
𝐁𝟏 = √𝐁𝐜𝟏 𝟐 + 𝐁𝐬𝟏 𝟐
In the same way:
𝐁𝟐 = √𝐁𝐜𝟐 𝟐 + 𝐁𝐬𝟐 𝟐
The angles θ1 and θ2 for the balancing masses B1 and B2 can be obtained as:
𝐁𝐬𝟏
𝛉𝟏 = 𝐭𝐚𝐧−𝟏 ( ) + 𝟏𝟖𝟎°
𝐁𝐜𝟏
𝐁𝐬𝟐
𝛉𝟐 = 𝐭𝐚𝐧−𝟏 ( ) + 𝟏𝟖𝟎°
𝐁𝐜𝟐
OBSERVATIONS AND CALCULATIONS:

Sr. m1 Distance Θd1 m2 Distance Θd2 b1 Distance b2 Distance Θ1 Θ2
No. (lb.) from P1 (deg.) (lb.) from P2 (deg.) (lb.) from P1 (lb.) from P1 (deg.) (deg.)
(in.) (in.) (in.) (in.)
1 1 0 10 1 13.5 10 2.11 7.50 0.11 21 190 10
2 1 0 90 1 13.5 90 2.11 7.50 0.11 21 270 90
3 2 0 180 2 13.5 180 2.11 7.50 0.22 21 360 180
COMMENTS:
 The apparatus is balanced without any added masses.
 The 2 disturbing masses are balanced by 2 rotating balancing
masses.
 Angles and radii should be carefully administered when
mounting the masses.
 Static as well as dynamic balance should be checked.
 Any vertical vibrations will mean that the balancing is not done
properly.
REFERENCES:
 Theory of Machines by S.S Rattan, Third Edition, Tata McGraw
Hill Education Private Limited.
 Kinematics and Dynamics of Machinery by R. L. Norton, First
Edition in SI units, Tata McGraw Hill Education Private Limited.
 “Primer on Dynamic Balancing “Causes, Corrections and
Consequences” By Jim Lyons International Sales Manager IRD
Balancing Div. EntekI-RD International
 Theory of Machines and Mechanisms SI Edition by Gordon R.
Pennock & Joseph E. Shigley John J. Uicker

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MECHANICS OF

DEPARTMENT OF MECHANICAL ENGINEERING

Why do we Balance components?

Main objective of Balancing:

Methods of Correcting Unbalance:

Static & Dynamic Unbalance:

Balancing of Rotating Masses:

Similarly, using the force balance along y axis:

Ms1 + Ms2 + Bs1 + Bs2 = 0

Bs1 (lB1 ) + Bs2 (lB2 ) + Ms2 (lM2 ) = 0

From Equation (1):

Putting value of Equation (5) in (3), we get:

[𝐌𝐜𝟏 + 𝐌𝐜𝟐 ](𝐥𝐁𝟏 )−𝐌𝐜𝟐 (𝐥𝐌𝟐 )

Using Equation (1), we have:

[𝐌𝐬𝟏 + 𝐌𝐬𝟐 ](𝐥𝐁𝟏 )−𝐌𝐬𝟐 (𝐥𝐌𝟐 )

𝐁𝐬𝟏 = −(𝐌𝐬𝟏 + 𝐌𝐬𝟐 +𝐁𝐬𝟐 )

In the same way:

OBSERVATIONS AND CALCULATIONS:

2 1 0 90 1 13.5 90 2.11 7.50 0.11 21 270 90

3 2 0 180 2 13.5 180 2.11 7.50 0.22 21 360 180

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