Vibration

1. Mechanical Vibrations
MEC4110
Unit I
Dr. Arshad Hussain Khan
Department of Mechanical Engineering
Zakir Husain College of Engg. & Technology
Aligarh Muslim University, Aligarh

2. Contents
• Mass or Inertia Element
• Combination of Masses
• Equivalent Mass
• Damping Elements
Lecture 3

3. Mass or Inertia Elements
 A mathematical model represents the actual
vibrating system, and there are often several
possible models.
 The purpose of the analysis generally determines
which mathematical model is appropriate.
 For eg. In case of modelling a multi-storey building
subjected to an earthquake, one possibility is that
the mass of the frame may be neglected compared
to the masses of the floors.
 The masses at the various floor levels represent the
mass elements, and the elasticities of the vertical
members denote the spring elements.

4. Combination of Massess
Several masses appear in combination in many practical applications. For
simplicity, we can replace these masses by a single equivalent mass.
 Translational Masses Connected by a Rigid Bar
Equivalent mass can be assumed to be located at any
point along the bar.
For e.g. if we wish to convert the three mass system
into a single mass system with equivalent mass at
location A.
From the equivalence of the kinetic energy of the three-
mass system to that of the equivalent single mass
system.

5.  Translational and Rotational Masses
Coupled Together
The two masses can be combined into either:
(1) A single equivalent translational mass, or
(2) A single equivalent rotational mass
Equivalent translational mass
kinetic energy of the original system (two masses)
kinetic energy of the equivalent system (Translational single mass)
Equivalence of kinetic energy
Equivalent rotational mass
kinetic energy of the original system (two masses)
kinetic energy of the equivalent system (Rotational single mass)

6. Ques: A cam-follower mechanism is used to convert the
rotary motion of a shaft into the oscillating or reciprocating
motion of a valve. The follower system consists of a pushrod
of mass mp rocker arm of mass mr and mass moment of
inertia Jr about its C.G., a valve of mass mv and a valve
spring of negligible mass. Find the equivalent mass meq. of
this cam-follower system by assuming the location of meq. as
(i) point A and (ii) point C.
Equivalent Mass
Sol.: Corresponding to displacement x= xp of the pushrod,
the rocker arm rotates by an angle about the
pivot point,
the valve moves downward by
and the C.G. of the rocker arm moves downward by
kinetic energy of the original system:
Case 1 If equivalent mass “meq.” is placed at point A ( )
Equivalence of kinetic energy
Case 2 If equivalent mass “meq.” is placed at point C ( )
Equivalence of kinetic energy

7. Equivalent Stiffness and Mass
Sol.:
Determine keq and meq for the system, when x, the
displacement of the center of the disk measured from
equilibrium, is used as the generalized coordinate,
Assume the disk is thin and rolls without slip.
Thin Disc
Pulley

8. Equivalent Mass
Sol.:
Find the equivalent mass “meq” of the system shown
in the Figure.
Total kinetic energy of the original system:
For sphere: Mass M.I. = 2/5 ms rs
2
Equivalence of kinetic energy

9. Damping Element
Damping The mechanism by which the vibrational energy is gradually converted into
heat or sound.
A damper is assumed to have neither mass nor elasticity, and damping force exists only if
there is relative velocity between the two ends of the damper.
Damping is modelled as one or more of the following types:
• Viscous Damping: When mechanical systems vibrate in a fluid medium such as air, gas, water, or oil, the
resistance offered by the fluid to the moving body causes energy to be dissipated. In viscous damping, the
damping force is proportional to the velocity of the vibrating body. Typical examples include (1) fluid film
between sliding surfaces, (2) fluid flow around a piston in a cylinder, (3) fluid flow through an orifice, and
(4) fluid film around a journal in a bearing.
• Coulomb or Dry-Friction Damping: Damping force is constant in magnitude but opposite in direction to
that of the motion of the vibrating body. It is caused by friction between rubbing surfaces that either are
dry or have insufficient lubrication.
• Material or Solid or Hysteretic Damping: When a material is deformed, energy is absorbed and
dissipated by the material due to friction between the internal planes, which slip or slide as the
deformations take place. When a body having material damping is subjected to vibration, the stress-strain
diagram shows a hysteresis loop. The area of this loop denotes the energy lost per unit volume of the
body per cycle due to damping

10. Thank you and
queries?