This document contains lecture notes on mechanical vibrations. It covers topics such as two degree of freedom systems, principal modes, double pendulums, torsional systems with damping, coupled systems, vibration absorbers, centrifugal pendulum absorbers, vibration isolators, and dampers. Examples of two degree of freedom systems include two masses connected by springs. Equations of motion are derived using mass and stiffness matrices. Torsional vibrations in shafts can be caused by inertia forces or shock loads. Centrifugal pendulum absorbers have a natural frequency that varies with rotational speed, making them well-suited for applications like engines.

1. Mechanical Engineering
COURSE NAME: MECHANICAL VIBRATIONS
Prepared By:
MD ATEEQUE KHAN
(Assistant Professor)
Mechanical Engineering Department
JIT,Barabanki,U.P. INDIA
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2. Table of Contents
Unit-3
1. Two Degree Freedom systems Introduction
2. Principal modes
3. Double pendulum
4. Torsional system with damping
5. Coupled system
6. Principle of vibration absorber
7. Undamped dynamic vibration absorbers
8. Torsional vibration absorber
9. Centrifugal pendulum absorbers
10.Vibration isolators .
11. Dampers.
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3. Unit-3.: Mechanical Vibrations
Two Degree Freedom systems Introduction
Introduction to two degree of freedom systems :
1. The vibrating systems, which require two coordinates to describe its motion, are called
two-degrees-of -freedom systems.
2.
These coordinates are called generalized coordinates when they are independent of each
other and equal in number to the degrees of freedom of the system.
3.
Unlike single degree of freedom system, where only one co-ordinate and hence one
equation of motion is required to express the vibration of the system, in two-dof systems
minimum two co-ordinates and hence two equations of motion are required to represent the
motion of the system. For a conservative natural system, these equations can be written by
using mass and stiffness matrices.
4.
One may find a number of generalized co-ordinate systems to represent the motion of the
same system. While using these co-ordinates the mass and stiffness matrices may be
coupled or uncoupled. When the mass matrix is coupled, the system is said to be
dynamically coupled and when the stiffness matrix is coupled, the system is known to be
statically coupled.
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4. Unit-3: Mechanical Vibrations
• Some examples of two-degree-of-freedom
systems :
• Figure shows two masses m1 and m2 with three springs having spring stiffness k1, k2 and k3 free to move on
the horizontal surface. Let x1 and x2 be the displacement of mass respectively:
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5. Unit-3: Mechanical Vibrations
FBD
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6. Unit-3: Mechanical Vibrations
Equation of motion in matrix form:
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7. Unit-3: Mechanical Vibrations
Double pendulum
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8. Unit-3: Mechanical Vibrations
Torsional system with damping
Torsional vibrations may result in shafts from following forcing:
1. Inertia forces of reciprocating mechanisms (such as pistons in Internal Combustion engines)
2. Impulsive loads occurring during a normal machine cycle (e.g. during operations of a punch press)
3. Shock loads applied to electrical machineries (such as a generator line fault followed by fault
removal and automatic closure)
4. Torques related to gear tooth meshing frequencies, turbine blade passing frequencies, etc.
5. For machines having massive rotors and flexible shafts (where system natural frequencies of
Torsional vibrations may be close to, or within, the source frequency range during normal
operation) torsional vibrations constitute a potential design problem area.
6. In such cases designers should ensure the accurate prediction of machine torsional frequencies and
frequencies of any of the torsional load fluctuations should not coincide with torsional natural
frequencies.
7. Hence, determination of torsional natural frequencies of a dynamic system is very important
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9. Unit-3: Mechanical Vibrations
Torsional system with damping
Fixed end
tk
pI
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10. Unit-3: Mechanical Vibrations
Coupled system
frictionless bearing
1PI 2PI
1
2
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11. Unit-3: Mechanical Vibrations
Coupled system
FBD
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12. Unit-3: Mechanical Vibrations
Torsional Mode
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13. Unit-3: Mechanical Vibrations
Governing Equation
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14. Unit-3: Mechanical Vibrations
Principle of vibration absorber
The centrifugal pendulum vibration absorber was devised and patented in
France about 1935 and at the same time it was independently conceived
and put into practice by E. S. Taylor. Its purpose was to overcome serious
torsional vibration problem inherent in geared radial aircraft-engine –
propeller system. Later it was modified and incorporated into automobile
IC engines in order to reduce the torsional vibrations of the crankshaft. This
was done by integrating the absorber mass with crankshaft counter balance
mass.
The tuned vibration absorber is only effective when the frequency of external
excitation equals to the natural frequency of the secondary spring and mass
system. But in many cases, for example in case of an automobile engine,
the exciting torques are proportional to the rotational speed ‘n' which may
vary over a wide range. For the absorber to be effective, its natural
frequency must also be proportional to the speed. The characteristics of the
centrifugal pendulum are ideally suited for this purpose.
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15. Unit-3: Mechanical Vibrations
Absorber
Tuned Vibration Absorber
Consider a vibrating system of mass , stiffness , subjected to a force . As studied in
case of forced vibration of single-degree of freedom system, the system will
have a steady state response which will be maximum when absorb this
vibration, one may add a secondary spring and mass system as shown in figure:
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16. Unit-3: Mechanical Vibrations
Centrifugal pendulum absorbers
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17. Unit-3: Mechanical Vibrations
Vibration isolators
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18. Unit-3: Mechanical Vibrations
Damper
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