MEC3120: Machinery Dynamics
Balancing
(Part-I)
Dr. M. Yaqoob Yasin
Email: yaqoob.yasin@gmail.com
DEPARTMENT OF MECHANICAL ENGINEERING
AMU ALIGARH
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Outline
Static and dynamic balancing of revolving
masses in one and different plane,
Balancing of reciprocating masses
Balancing of V-Engine, in–line and radial I.C.
Engine.
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A net unbalanced force acting on the frame of a machine or
mechanism (i.e., the resultant of the forces transmitted at all of the
connections between the machine and frame) is referred to as a
shaking force.
Likewise, a resultant unbalanced moment acting on the frame is called
a shaking moment.
Since the shaking force and shaking moment are unbalanced effects,
they will cause the frame to vibrate, with the magnitude of the
vibration dependent on the amount of unbalance.
Thus, an important design objective is to minimize machine unbalance.
Often the unbalance of forces is produced in rotary or reciprocating
machinery due to the inertia forces associated with the moving masses.
Balancing is the process of designing or modifying machinery so that
the unbalance is reduced to an acceptable level and if possible is
eliminated entirely.
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INTRODUCTION
The method of balancing that is employed depends to a considerable
extent on the type of unbalance present in the machine.
The two basic types are rotating unbalance and reciprocating
unbalance, which may occur separately or in combination.
Balancing procedures for sizing and positioning corrective masses on rotors are
based on the following criteria:
1. For static balance, the shaking force must be zero.
2. For dynamic balance, the shaking force and the shaking moment must both be
zero.
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Continue…
In general, static unbalance is characterized by a net shaking force, and dynamic
unbalance is characterized by a combination of a net shaking force and a net
shaking couple.
Dynamic unbalance is more apt to be significant in cases of rotors having their mass
distributed over relatively large axial distances. For example, static balancing may
be satisfactory for machine components such as automobile wheels or household
window fans, which have short axial lengths, whereas dynamic balancing must be
performed on equipment such as automotive crankshafts and multistage turbine
rotors that have large axial lengths.
Most of the serious problems encountered in high-speed machinery are the direct
result of Unbalanced forces. These forces exerted on the frame by the moving
machine members are time varying, impart vibratory motion to the frame and
produce noise. Also, there are human discomfort and detrimental effects on the
machine performance and the structural integrity of the machine foundation.
The most common approach to balancing is by redistributing the mass which may
be accomplished by addition or removal of mass from various machine members.
There are two basic type of unbalance- Rotating unbalance and Reciprocating
unbalance- which may occur separately or in combination.
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Static balancing
A system of rotating masses is said to be in static balance if the combined mass centre of the
system lies on the axis of rotation.
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(1)
If this vector sum is zero, then the rotor is balanced.
The magnitude and location of this counterweight are determined from the condition that
the resultant inertial force must now be zero; that is,
(2)
(3)
In general, for N initial masses, the balancing condition is
(4)
Since all the vectors in Eq. (3) lie in a plane that is parallel to the yz-plane in figure 2 (a), that
equation is a two-dimensional vector equation. Equation (3) can also be solved
mathematically by dividing it into y and z components. We get
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(5)
and
(6)
(7)
and
(8)
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Sample Problem on static balancing of a Rotor
The rigid rotor has the following
properties:
Determine the amount and location of the counterweight required for static balance.
Solution. Substituting the values in given Eqn.
This product will result, for example, from a counterweight mass of 2.85 kg at a
radial distance of 80 mm.
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The angular position of the counterweight is calculated as
where from the signs of the numerator and denominator in the argument of the
arctan function indicate that the angle is in the fourth quadrant.
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Dynamic Balancing
Figure 3(a) shows a rotor with eccentric masses at multiple axial locations. As a result, the
rotor experiences general dynamic unbalance.
For static balance, the sum of all
the inertial forces must be zero, a
condition that yields the following
equations,
For the general case of N original masses,
we have
Fig3. (a) In general, dynamic balancing requires the use of two
counterweights. Shown are counterbalances placed in arbitrarily selected
planes at axial positions P and Q. (b) Graphical determination of
counterweight 2. (c) Graphical determination of counterweight 1.
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However, a shaking couple will still exist if the inertial forces produce a net couple. Therefore,
the additional condition for dynamic balance is that the sum of the moments of the inertia forces
about any arbitrary point be zero.
For convenience in determining the required counterbalances, we will take moments about point
P, the axial location of counterweight 1, thereby eliminating this unknown counterweight from
the moment equation.
The axial distances of all other masses relative to point P are designated by symbol s.
Taking the sum of the inertial force moments about point P is
we can rearrange this equation as
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Equation is resolved into component form:
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Sample Problem on dynamic balancing of a Rotor
The rigid rotor has the following properties:
The total axial length is 1000 mm between bearings. Counterweights are to be placed in planes
that are 100 mm from each bearing. The axial distances is shown in figure,
Determine the amounts and
locations of the counterweights in
planes P and Q required for
complete balance.
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THANK YOU
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