Dr. S. K. Kudari,

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Dr. S. K. Kudari, Professor, Department of
Mechanical Engineering, B V B College of Engg.
Tech., HUBLI B.E.(Mech), Karnataka University,
Dharwad M.Tech. (Machine Design), IIT,
Kharagpur Ph.D. (Mechanical Engg), IIT,
Kharagpur 19 Years of teaching experience 28
Publications Co-author of special Indian edition
of the book Mechanical vibrations by Kelly,
Tata McGraw hill Publications.
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CHAPTER-5
VIBRATION MEASURING INSTRUMENTS
Topics covered Theory of Vibration measuring
instruments Displacement measuring instrument
(Vibrometer) Velocity measuring instrument
(Velometer) Acceleration measuring instrument
(Accelerometer) Numerical Problems/Discussions
03/04/07
04/04/07
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
3
CHAPTER-5
Topics covered Whirling of shafts neglecting
damping Whirling of shafts with damping
Numerical Problems/Discussions
10/04/07
11/04/07
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
4
Theory of Vibration measuring instruments
Introduction
The dynamic forces in a vibratory systems depend
on the displacement, velocity and acceleration
components of a system
Spring force ? displacement Damping force ?
velocity Inertia force ? acceleration
Machine
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
5
Theory of Vibration measuring instruments
Introduction
Therefore, In vibration analysis of a mechanical
system, it is required to measure the
displacement, velocity and acceleration
components of a system
Machine
An instrument, which is used to measure these
parameters, is referred as vibration measuring
instrument or seismic instrument
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Theory of Vibration measuring instruments
Introduction
The major requirement of a seismic instrument is
to indicate an output, which represents an input
such as the displacement amplitude, velocity or
acceleration of a vibrating system as close as
possible.
Machine
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
7
Theory of Vibration measuring instruments
m-seismic mass c-damping coefficient of seismic
unit K-stiffness of spring used in seismic
unit x-absolute displacement of seismic
mass y-base excitation (assume SHM) z(x-y)
displacement of seismic mass relative to frame
Model
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Theory of Vibration measuring instruments
Dynamic
Static
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Theory of Vibration measuring instruments
Equation of motion of the seismic mass
..(1)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Theory of Vibration measuring instruments
The governing equation of motion of the system is
(4)
Solution of governing differential equation
................(5)
Transient solution Steady state solution
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Theory of Vibration measuring instruments
Displacement
Differentiating
Velocity
Acceleration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Theory of Vibration measuring instruments
..(7)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Theory of Vibration measuring instruments
Vectorial representation of forces
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Theory of Vibration measuring instruments
From triangle OAB
..(9)
..(10)
.(11)
divide the above Eqn. by K
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Theory of Vibration measuring instruments
(12)
..(13)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Theory of Vibration measuring instruments
(14)
the phase angle (from force diagram) is
....(15)
OR
...(16)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Theory of Vibration measuring instruments
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Theory of Vibration measuring instruments
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Displacement measuring instrument (Vibrometer)
It is an instrument used to measure the
displacement of a vibrating system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Displacement measuring instrument (Vibrometer)
Condition for Vibrometer
To satisfy the above equation the frequency
ratio, r, must be large, i.e natural frequency of
vibrometer must be low compared to that of
vibration to be measured.
i.e ?n (K/m)1/2 must be low, it can be achieved
by higher mass and lower stiffness of the spring.
This condition results in bulky instrument.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Velocity measuring instrument (Velometer)
It is an instrument used to measure the velocity
of a vibrating system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Velocity measuring instrument (Velometer)
The seismic response is
velocity component of the seismic response is
. .(21)
velocity component of the machine vibration is
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Velocity measuring instrument (Velometer)
Condition for Velometer
.(22)
With some phase lag ?
Then
(23)
The seismic instrument can be calibrated so that
the record directly gives value of velocity of
base excitation
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
24
Acceleration measuring instrument (Accelerometer)
It is an instrument used to measure the
acceleration of a vibrating system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Acceleration measuring instrument (Accelerometer)
The seismic response is
where
Acceleration component of the response is
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Acceleration measuring instrument (Accelerometer)
(24)
(25)
(26)
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Acceleration measuring instrument (Accelerometer)
Condition for Accelerometer
If
(27)
...(28)
Then,
Acceleration of the machine is
(29)
acceleration of the base with phase lag ?
The seismic instrument can be calibrated so that
the record directly gives value of acceleration
of base excitation
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
28
Acceleration measuring instrument (Accelerometer)
For accelerometer, the frequency ratio, r, must
be very small between 0-0.6
Since, r is small for accelerometer, the natural
frequency of vibration should be high compared to
the frequency of vibration of base.
As , for condition to be
satisfied seismic mass should be small and spring
stiffness should be high. This indicates this
instrument will be small in size
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Seismic instrument
0.707
Is more suited for seismic instruments
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
30
Summary
Seismic instruments are used to measure the
displacement, velocity and acceleration
components of a vibratory system
Basic theory of Seismic instruments is based on
forced vibration considering the vibratory system
under base excitation.
A single Seismic instrument can be sued as
vibrometer, velometrer and accelerometer using
suitable calibration.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
31
Thank You
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli