Machine Dynamics and Vibration
Dr. Syed Mamun R Rasid
January 30, 2025
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
1 / 38
Gear Trains
Gear Train
Sometimes, two or more gears are made to mesh with each other to
transmit power from one shaft to another. Such a combination is called
gear train or train of toothed wheels. The nature of the train used depends
upon the velocity ratio required and the relative position of the axes of
shafts. A gear train may consist of spur, bevel or spiral gears.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
2 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
1
Simple gear train
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
1
Simple gear train
2
Compound gear train
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
1
Simple gear train
2
Compound gear train
3
Reverted gear train
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
1
Simple gear train
2
Compound gear train
3
Reverted gear train
4
Epicyclic gear train
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Types of Gear Trains
Problem Solving
The following are the main types of gear trains:
1
Simple gear train
2
Compound gear train
3
Reverted gear train
4
Epicyclic gear train
In the first three types of gear trains, the axes of the shafts over which the
gears are mounted are fixed relative to each other. But in case of epicyclic
gear trains, the axes of the shafts on which the gears are mounted may
move relative to a fixed axis.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
3 / 38
Gear Train
Simple Gear Train
A series of gears,
capable of receiving
and transmitting
motion from one gear
to another is called a
simple gear train. In
it, all the gear axes
remain fixed relative
to the frame and each
gear is on a separate
shaft.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
4 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
1
Two external gears of a pair always move in opposite directions.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
1
2
Two external gears of a pair always move in opposite directions.
All odd-numbered gears move in one direction and all even-numbered
gears in the opposite direction.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
1
2
3
Two external gears of a pair always move in opposite directions.
All odd-numbered gears move in one direction and all even-numbered
gears in the opposite direction.
Speed ratio (or velocity ratio) of gear train is the ratio of the speed of
the driver to the speed of the driven or follower and ratio of speeds of
any pair of gears in mesh is the inverse of their number of teeth
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
1
2
3
4
Two external gears of a pair always move in opposite directions.
All odd-numbered gears move in one direction and all even-numbered
gears in the opposite direction.
Speed ratio (or velocity ratio) of gear train is the ratio of the speed of
the driver to the speed of the driven or follower and ratio of speeds of
any pair of gears in mesh is the inverse of their number of teeth
The ratio of the speed of the driven or follower to the speed of the
driver is known as train value of the gear train
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Simple Gear Train
Important Notes for Simple Gear Train:
1
2
3
4
5
Two external gears of a pair always move in opposite directions.
All odd-numbered gears move in one direction and all even-numbered
gears in the opposite direction.
Speed ratio (or velocity ratio) of gear train is the ratio of the speed of
the driver to the speed of the driven or follower and ratio of speeds of
any pair of gears in mesh is the inverse of their number of teeth
The ratio of the speed of the driven or follower to the speed of the
driver is known as train value of the gear train
Intermediate gears have no effect on the speed ratio and, therefore,
they are known as idlers.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
5 / 38
Gear Train
Compound Gear Train
Compound Gear Train
When a series of gears are connected in such
a way that two or more gears rotate about
an axis with the same angular velocity, it is
known as compound gear train. In this type,
some of the intermediate shafts, i.e., other
than the input and the output shafts, carry
more than one gear.
whenever the distance between the driver
and the driven or follower has to be bridged
over by intermediate gears and at the same
time a great ( or much less ) speed ratio is
required, then the advantage of intermediate
gears is intensified by providing compound
gears on intermediate shafts.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
6 / 38
Gear Train
Advantage of Compound Gear Train
Advantage of Compound Gear Train
The advantage of a compound train over a simple gear train is that a
much larger speed reduction from the first shaft to the last shaft can be
obtained with small gears. If a simple gear train is used to give a large
speed reduction, the last gear has to be very large. Usually for a speed
reduction in excess of 7 to 1, a simple train is not used and a compound
train or worm gearing is employed.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
7 / 38
Gear Train
Design of Spur Gears
The spur gears (i.e. driver and driven) are to be designed for the given
velocity ratio and distance between the centres of their shafts.
Let x = Distance between the centres of two shafts,
N1 = Speed of the driver,
T1 = Number of teeth on the driver,
d1 = Pitch circle diameter of the driver,
N2 , T2 and d2 = Corresponding values for the driven or follower, and
pc = Circular pitch.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
8 / 38
Gear Train
Design of Spur Gears
The distance between the centres of two shafts,
x=
d1 + d2
2
and speed ratio or velocity ratio,
N1 d 2 T 2
=
=
N2 d 1 T 1
From the above equations, we can conveniently find out the values of d1
and d2 (or T1 and T2 ) and the circular pitch ( pc ).
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
9 / 38
Gear Train
Reverted Gear Train
Reverted Gear Train
When the axes of the first gear (i.e.
first driver) and the last gear (i.e.
last driven or follower) are co-axial,
then the gear train is known as
reverted gear train. In a reverted
gear train, the motion of the first
gear and the last gear is like.
The reverted gear trains are used in
automotive transmissions, lathe back
gears, industrial speed reducers, and
in clocks (where the minute and hour
hand shafts are co-axial).
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
10 / 38
Gear Train
Epicyclic Gear Train
Epicyclic Gear Train
A gear train having a relative motion of axes is
called a planetary or an epicyclic gear train (or
simply epicyclic gear or train). In an epicyclic
train, the axis of at least one of the gears also
moves relative to the frame.
Consider two gear wheels S and P, the axes of
which are connected by an arm a. If the arm a is
fixed, the wheels Sand P constitute a simple
train. However, if the wheel Sis fixed so that the
arm can rotate about the axis of S, the wheel P
would also move around S. Therefore, it is an
epicyclic train. The epicyclic gear trains may be
simple or compound.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
11 / 38
Gear Train
Epicyclic Gear Train
Advantages of Epicyclic Gear
Train
The epicyclic gear trains are useful for
transmitting high velocity ratios with gears of
moderate size in a comparatively lesser space.
Application of Epicyclic Gear
Train
The epicyclic gear trains are used in the back
gear of lathe, differential gears of the
automobiles, hoists, pulley blocks, wrist watches
etc.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
12 / 38
Gear Train
Velocity Ratio of Epicyclic Gear Train
The following two methods may be used for finding out the velocity ratio
of an epicyclic gear train.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
13 / 38
Gear Train
Velocity Ratio of Epicyclic Gear Train
The following two methods may be used for finding out the velocity ratio
of an epicyclic gear train.
1
Tabular method
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
13 / 38
Gear Train
Velocity Ratio of Epicyclic Gear Train
The following two methods may be used for finding out the velocity ratio
of an epicyclic gear train.
1
Tabular method
2
Algebraic method.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
13 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Ts
x
revolutions made by P = − T
p
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Ts
x
revolutions made by P = − T
p
Let the locked system be turned through y
revolutions in the clockwise direction. Then
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Ts
x
revolutions made by P = − T
p
Let the locked system be turned through y
revolutions in the clockwise direction. Then
revolutions made by a = y
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Ts
x
revolutions made by P = − T
p
Let the locked system be turned through y
revolutions in the clockwise direction. Then
revolutions made by a = y
revolutions made by S = x + y
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Tabular Method
Assume that the arm a is fixed. Turn S through x
revolutions in the clockwise direction. Assuming
clockwise motion of a wheel as positive and
counter-clockwise as negative.
revolutions made by a = 0 (arm is fixed)
revolutions made by S = x
Ts
x
revolutions made by P = − T
p
Let the locked system be turned through y
revolutions in the clockwise direction. Then
revolutions made by a = y
revolutions made by S = x + y
Ts
revolutions made by P = y − T
x
p
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
14 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
15 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Lock the arm and assume the other wheels free to rotate.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
15 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Lock the arm and assume the other wheels free to rotate.
Tum any convenient gear through one revolution in the clockwise
direction and record the number of revolutions made by each of the
other wheels.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
15 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Lock the arm and assume the other wheels free to rotate.
Tum any convenient gear through one revolution in the clockwise
direction and record the number of revolutions made by each of the
other wheels.
Multiply all the above recordings by x and write the same in the
second row. This is equivalent to the statement that the chosen wheel
is given x revolutions in the clockwise direction keeping the arm fixed.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
15 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Lock the arm and assume the other wheels free to rotate.
Tum any convenient gear through one revolution in the clockwise
direction and record the number of revolutions made by each of the
other wheels.
Multiply all the above recordings by x and write the same in the
second row. This is equivalent to the statement that the chosen wheel
is given x revolutions in the clockwise direction keeping the arm fixed.
Add y to all the quantities in the second row and make the recordings
in the third row. This amounts to the fact that by locking the whole
system, it is turned throughy revolutions in the clockwise direction.
Thus, the arm makes y revolutions, the chosen wheel (y + x)
revolutions, and so on.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
15 / 38
Gear Train
Summary of Tabular Method
The procedure can be summarised as follows:
Lock the arm and assume the other wheels free to rotate.
Tum any convenient gear through one revolution in the clockwise
direction and record the number of revolutions made by each of the
other wheels.
Multiply all the above recordings by x and write the same in the
second row. This is equivalent to the statement that the chosen wheel
is given x revolutions in the clockwise direction keeping the arm fixed.
Add y to all the quantities in the second row and make the recordings
in the third row. This amounts to the fact that by locking the whole
system, it is turned throughy revolutions in the clockwise direction.
Thus, the arm makes y revolutions, the chosen wheel (y + x)
revolutions, and so on.
Machine and
Dynamics
and Vibration
30, 2025
Appl y the given conditions
find
the values of x January
and y.
Having 15 / 38
Dr. Syed Mamun R Rasid
Gear Train
Summary of Tabular Method
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
16 / 38
Gear Train
Algebric Method
Relative Velocity Method
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
or Ns = Nsa + Na
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
or Ns = Nsa + Na
Simillarly, NP = −Npa + Na
Nsa = Ns − Na
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
or Ns = Nsa + Na
Simillarly, NP = −Npa + Na
Nsa = Ns − Na
Npa = Na − Np
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
or Ns = Nsa + Na
Simillarly, NP = −Npa + Na
Nsa = Ns − Na
Npa = Na − Np
Nsa
Ns − Na
=
Npa
Na − Np
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Algebric Method
Relative Velocity Method Angular vel. of S = angular vel. of S rel. to a +
angular vel. of a
or ωs = ωsa + ωa
or Ns = Nsa + Na
Simillarly, NP = −Npa + Na
Nsa = Ns − Na
Npa = Na − Np
Nsa
Ns − Na
=
Npa
Na − Np
Tp
Ns − Na
=
Ts
Na − Np
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
17 / 38
Gear Train
Compound Epicyclic Gear Train
Compound Epicyclic Gear Train—Sun and Planet Gear
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
18 / 38
Gear Train
Compound Epicyclic Gear Train
Compound Epicyclic Gear Train—Sun and Planet Gear A compound epicyclic
gear train consists of two co-axial shafts S1 and S2 , an annulus gear A which
is fixed, the compound gear (or planet gear) B-C, the sun gear D and the
arm H. The annulus gear has internal teeth and the compound gear is carried by the arm and revolves freely on a pin of the arm H. The sun gear is
co-axial with the annulus gear and the arm but independent of them. The
gear at the centre is called the sun gear and the gears whose axes move are
called planet gears.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
18 / 38
Gear Train
Problem Solving
Practice Problem: In an epicyclic gear train, the internal wheels A and
B and compound wheels C and D rotate independently about axis O. The
wheels E and F rotate on pins fixed to the arm G. E gears with A and C
and F gears with B and D. All the wheels have the same module and the
number of teeth are: TC = 28; TD = 26; TE = TF = 18. 1. Sketch the
arrangement ; 2. Find the number of teeth on A and B; 3. If the arm G
makes 100 r.p.m. clockwise and A is fixed, find the speed of B; and 4. If the
arm G makes 100 r.p.m. clockwise and wheel A makes 10 r.p.m. counter
clockwise; find the speed of wheel B.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
19 / 38
Balancing
What is Balancing?
1
2
Moving parts of a machine will have associated (i) applied force and
torque (ii) inertia forces/inertia torques
Inertia forces and/or inertia torques primary cause primarily cause the
unbalance in the rotary and reciprocating parts.
Inertia Forces = Shaking or rocking or unbalanced or disturbing forces
Inertia Torques = Shaking or rocking or unbalanced or disturbing
Definition (Balancing)
Balancing is the process of correcting or eliminating unbalance due to
inertia forces and moments.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
20 / 38
Balancing
Introduction
Unbalanced Force Due to Revolving Rotor
Figure 1: Unbalance Force
Unbalace force = Inertia force = Centrifugal force
The most common approach to balancing
1
By adding balancing mass
2
By removing mass from the machine member.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
21 / 38
Balancing
Importance of Balancing
1
If the moving parts are not balanced
2
Inertia force/moments are set up in the structure
3
Induced vibration in the structure
1
Produce excessive noise
2
cause undue wear in the mating components
3
increase the stress in the component in
This will lead to
premature failure of the machine components
faulty performance of the machine
human discomfort
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
22 / 38
Balancing
Types of Mass Balancing Problem
1
2
Balacing of rotating masses
1
Balancing of single rotating mass
2
Balancing of several masses rotating in the same plane
3
Balancing of several masses rotating in the different plane
Balacing of reciprocating masses
1
Balancing of single cylinder engines
2
Balancing of two cylinder engines
3
Balancing of multi-cylinder in-line engines
4
Balancing of radial engines
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
23 / 38
Balancing
Balancing of Rotating Masses
Balancing Single Rotating Mass
Figure 2: Caption
Unbalanced force = Inertia force = Centrifugal force = Fc = mω 2 r
There are two ways to balance this unbalance force:
1
By introducing single revolving mass in the same plane
2
By introducing two revolving masses in the different plane
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
24 / 38
Balancing
Balancing Single Rotating Mass
Method 1: By introducing single revolving mass in the same plane
Figure 3: Caption
Disturbing Force Fc1 = mω 2 r
Balancing Force Fc2 = mB ω 2 rB
But for Balancing Fc1 = Fc2
Dr. Syed Mamun R Rasid
mr = mB rB
Machine Dynamics and Vibration
January 30, 2025
25 / 38
Balancing
Balancing Single Rotating Mass
Method 2: By introducing two revolving masses in the different plane
If the disturbing mass and
balancing mass lie in different
plane, disturbing mass can
not be balanced by single
mass as there will be a couple
left unbalanced. So it
requires at least two
balancing masses.
Condition for Balancing:
The resultant centrifugal
force must be zero.
The resultant couple
must be zero.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
26 / 38
Balancing
Balancing Single Rotating Mass
Method 2: By introducing two revolving masses in the different plane
Applying Condition 1 of Force
Balancing:
mr = mB1 rB1 + mB2 rB2
Applying condition 2 of
Couple Balancing:
mB1 rB1 l = mrl2
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
27 / 38
Balancing
Balancing Several Rotating Mass
Consider a four rotor system rotating in the same plane:
Given Data:
mass m1 , m2 , m3 , m4
radius of rotation
r1 , r2 , r3 , r4
relative angular position
θ1 , θ 2 , θ 3 , θ 4
To Find:
Magnitude of balancing
masses mB
Angular Position of
balancing masses θB
relative angular position
θ1 , θ 2 , θ 3 , θ 4
Solving Method:
Analytical Method
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
28 / 38
Balancing
Balancing Several Rotating Mass
Method 1: Analytical
Step 1: Find centrifugal forces
exerted by each rotating masses.
Step 2: Resolve the centrifugal
forces horizontally and vertically
and find their sums.
Step 3: Find magnitude and
direction
resultant force.
q of
P
P
FCR = ( FH )2 + ( FV )2
P
F
θR = tan1 ( P FV )
H
Find magnitude and direction of
balancing mass FB = mB rB
θB = θR + 180
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
29 / 38
Balancing
Balancing Several Rotating Mass
Method 2: Graphical
1
2
Step 1: Draw the space diagram
with the positions of the several
masses.
Step 2: Find centrifugal force
exerted by each rotating mass
Step 3: Draw force polygon.
If force polygon is closed, there
is no out of balance force.
If force polygon is open, there is
out of balance force.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
30 / 38
Balancing
Balancing Several Rotating Mass
Method 2: Graphical
Step 1: Draw the space diagram
with the positions of the several
masses.
Step 2: Find centrifugal force
exerted by each rotating mass
Step 3: Draw force polygon.
1 If force polygon is closed, there
is no out of balance force.
2 If force polygon is open, there is
out of balance force.
Resultant force = closing side of the
polygon.
Balancing force = Resultant force &
in opposite direction.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
31 / 38
Balancing
Balancing Several Mass Revolving in different plane
Method 2: Graphical
When the system rotates
without any vibration, that
means the system is in dynamic
balance.
When the system vibrates
during rotation, that means the
system is out of balance. It
requires dynamic balancing.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
32 / 38
Balancing
Balancing Several Mass Revolving in different plane
DALBY’s Method
The technique of tackling this
types of problem is to transfer
the centrifugal force acting in
each plane to a single parallel
plane which is usually termed as
reference plance.
Then the procedure for
balancing is almost the same as
for balancing several masses
rotating in the same plane.
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
33 / 38
Balancing
Balancing Several Mass Revolving in different plane
Transfer of a force from one plane to another plane
Figure 4: Fig. 132.pdf
1
2
3
The effect of transferring a force F acting in one plane to another
plane (reference plane) is equivalent to transfer the same force F in
magnitude and direction in the reference plane accompanied by a
couple of magnitude F × l.
Couple vector is perpendicular (90 degrees) to force vector
In Balacing problems, it is convenient if couple vectors are drawn by
turning them through 90 degrees in the same sense (by drawing them
toR Rasid
the force vector).
This does
not affect their
relative
Dr.parallel
Syed Mamun
Machine Dynamics
and Vibration
January
30, 2025
34 / 38
Balancing
Balancing of Reciprocating Masses
Acceleration of the reciprocating mass of a slider-crank mechanism is given
by
cos 2θ
a = ω 2 r [cos θ +
]
n
Therefore, the force required to accelerate mass m is
FI = FU = mω 2 r [cos θ +
cos 2θ
]
n
Primary accelerating force = mω 2 r cos θ
Secondary accelerating force = mω 2 r cosn2θ
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
35 / 38
Balancing
Partial Balancing of Unbalanced Primary Force in a Reciprocating
Engine
The primary unbalanced force
mω 2 r cos θ may be considered as the
component of the centrifugal force
produced by a rotating mass m
placed at the crank radius r. The
centrifugal force due to mass B =
Bω 2 b
Horizontal component of this force
acting in opposite direction of
primary force = Bω 2 b cos θ
The primary force is balanced, if
mω 2 r cos θ = Bω 2 b cos θ
mr = Bb
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
36 / 38
The
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
37 / 38
Balancing
Partial Balancing of Unbalanced Primary Force in a Reciprocating
Engine
The primary force is completely balanced if B.b = m.r, but the centrifugal
force produced due to the revolving mass B, has also a vertical component
(perpendicular to the line of stroke) of magnitude Bω 2 b sin θ. This force
remains unbalanced. The maximum value of this force is equal to Bω 2 b
when θ is 90° and 270°, which is same as the maximum value of the
primary force mω 2 r .
Figure 5: Caption
Dr. Syed Mamun R Rasid
Machine Dynamics and Vibration
January 30, 2025
37 / 38
Balancing
Partial Balancing of Unbalanced Primary Force in a Reciprocating
Engine
As a compromise let a fraction ‘c’ of the reciprocating masses is balanced,
such that c.m.r = B.b
Unbalanced force along the line of stroke
= mω 2 r cos θ − Bω 2 b cos θ =mω 2 r cos θ − cmω 2 r cos θ=(1 − c)mω 2 r cos θ
Unbalanced force along the perpendicular to the line of stroke
=Bω 2 r sin θ = cmω 2 r sin θ
Resultant unbalanced force at any instant
=
q
[(1 − c)mω 2 r cos θ]2 + [cmω 2 r sin θ]2
=mω 2 r
Dr. Syed Mamun R Rasid
q
(1 − c)2 cos2 θ + c 2 sin2 θ
Machine Dynamics and Vibration
January 30, 2025
38 / 38