This document discusses a project analyzing the vibrational behavior of a cantilever rotor with cracks rotating in a viscous medium. The objectives are to analyze the rotor dynamically both with and without cracks in various mediums using theoretical calculations, experimental testing, and established theories. Results show that the presence of a crack and more viscous mediums decrease the critical speed and increase the amplitude of vibration. Further analysis of bearing characteristics and gyroscopic effects are recommended.

1. VIBRATIONAL ANALYSIS OF CANTILEVER ROTOR
IN VISCOUS MEDIUM
Under the expert guidance and mentorship of
Prof. Isham Panigrahi
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SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY

2. INTRODUCTION-CRACKS IN A
SHAFT/BEAM
Presences of cracks in rotating shafts are serious threats to its
performance. detection of crack in rotors needs urgent
attention.
Precautions should be taken much earlier as crack propagates quicker
in rotating shafts due to fatigue loading.
Cracks are the major causes of failure, investigation for vibration
analysis of rotor with cracks are essential for safe design.Viscous
medium, the analysis of critical speed becomes complex.
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3. PRACTICAL APPLICATIONS OF CRACK
INVESTIGATION IN SHAFTS AND
BEAMS
The analysis of a cracked rotating shaft in viscous medium will be
utilized for
I )condition monitoring
II ) for early crack detection in rotor for vibration of
(a) high-speed rotor in centrifuges(prone to fatigue)
(b) high-speed boring machine
(c) rotors used for drilling oil from sea bed
III) preventing failure of rotors used in machineries subjected to
various environmental conditions.
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4. OBJECTIVES OF THE PROJECT
The phases of the process plan for the present Investigation are as follows:
• Dynamic analysis of cracked cantilever rotor without viscous medium.
• Dynamic analysis of cracked cantilever rotor in viscous medium.
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5. IMPORTANT THEORIES AND
METHODS USED FOR THE ANALYSIS
Cracks introduce new boundary conditions for the structures at the crack
locations. These boundary conditions are derived from the strain energy
equation using Castigliano’s theorem.
Presence of crack also reduces stiffness of the structures which has been
derived from stiffness matrix.
Euler-Bernoulli beam theory is used for dynamic characteristics of beams with
transverse cracks.
Timoshenko Beam theory is successfully used for vibration analysis of cracked
shaft.
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6. IMPORTANT THEORIES AND
METHODS USED FOR THE ANALYSIS
The dynamic response of rotors with transverse cracks rotating in viscous
medium, the amplitude of vibration of rotors are found using Navier-Stokes
equation and Fourier transform technique.
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7. APPARATUS REQUIRED
An electric motor
(203v,50hz,0.75amps,
1/8hp,93w,1350rpm)
A flexible coupling
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8. Bearing with housing
Dial indicator
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9. Rotor
Weight of rotor = 1557.5 grams
Length of the shaft of the
rotor = 400mm = 40cm
Diameter of the shaft of
the rotor = 20mm = 2cm
Thickness of the disc of
the rotor = 15.5mm=1.55cm
Diameter of the disc of the rotor
= 84mm = 8.40 cm
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10. Stroboscope
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11. EXPERIMENTAL SETUP
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12. THEORETICAL CALCULATION
APPROACH #1
Volume of the rotor = volume of the shaft + volume of the disc
=3.14*40*(2*2)/4+3.14*1.55*(8.40*8.40)/4
=211.6102cm3
density of the rotor = mass of the rotor / volume of the rotor
= 1557.5/211.6102
= 7.3602 g / cm3 = 7360.2 kg / m3
Theoretically, C = (Modulus of rigidity / density of the rotor ) 1/2
= (80*109/7360.2)1/2
= 3.29*103 m/s .
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13. APPROACH #1(CONTD.)
Again, wn = (2n + 1)*3.14*C) / 2*l
wn = 3.14* C / (81.55/100)
wn = 12.66*10 3 rad/s
Therefore the natural frequency is f n = 12.66*10 3 / 2*3.14
fn= 2015.92 Hz
Therefore the theoretical rpm at which the first frequency occurs
is = fn * 60
= 120955.4 rpm.
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14. APPROACH #2
Now following another approach to find out the natural frequency
Stiffness of any beam is given by K t = 3.14*G*d4 / 32*l
Kt = 3140
Polar moment of inertia of a beam is given by J 0 = density * height of the
rotor * 3.14 * (D) 4 / 32
J0 = 1.7 * 10-3
Now to find out wn = ( Kt / Jo )
wn = 1359.065 rad/sec .
Therefore the natural frequency is f n = Wn / 2*3.14
fn= 216.411 Hz
Therefore the theoretical rpm at which the first frequency occurs is = f n * 60
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= 12984.66 rpm.
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15. EXPERIMENTAL ANALYSIS
WITHOUT CRACK IN AIR
1.Deflection in air
At 1446 rpm
Deflection shown by the vibration meter = 0.05mm
Dial indicator reading = 0.1 mm
At 847 rpm
Deflection shown by the vibration meter = 0.95 mm
Dial indicator reading = 0.9 mm
At 331 rpm
Deflection shown by the vibration meter = 1.123 mm
Dial indicator reading = 1.2 mm
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16. EXPERIMENT ANALYSIS(CONTD.)
2 .Deflection in water
At 1446 rpm
Deflection shown by the vibration meter = 0.18 mm
Dial indicator reading = 0.2 mm
At 847 rpm
Deflection shown by the vibration meter = 0.78mm
Dial indicator reading = 0.7 mm
At 331 rpm
Deflection shown by the vibration meter = 1.87 mm
Dial indicator reading = 1.9 mm
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17. EXPERIMENTAL
ANALYSIS(CONTD.)
3. Deflection in flour-water
At 1446 rpm
Deflection shown by the vibration meter = 0.57 mm
Dial indicator reading = 0.6 mm
At 847 rpm
Deflection shown by the vibration meter = 1.12 mm
Dial indicator reading = 1.2 mm
At 331 rpm
Deflection shown by the vibration meter = 1.93 mm
Dial indicator reading = 1.9 mm
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18. EXPERIMENTAL
ANALYSIS(CONTD.)
WITH CRACK IN ROTOR
1. DEFLECTION IN AIR
At 1446 rpm
Deflection shown by the vibration meter = 0.06 mm
Dial indicator reading = 0.1 mm
At 847 rpm
Deflection shown by the vibration meter = 0.85 mm
Dial indicator reading = 0.9 mm
At 331 rpm
Deflection shown by the vibration meter = 1.341 mm
Dial indicator reading = 1.5 mm
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19. EXPERIMENTAL
ANALYSIS(CONTD.)
2.DEFLECTION IN WATER
At 1446 rpm
Deflection shown by the vibration meter = 0.23 mm
Dial indicator reading = 0.3 mm
At 847 rpm
Deflection shown by the vibration meter = 0.95 mm
Dial indicator reading = 0.9 mm
At 331 rpm
Deflection shown by the vibration meter = 1.8 mm
Dial indicator reading = 1.8 mm
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20. EXPERIMENTAL
ANALYSIS(CONTD.)
3. DEFLECTION IN FLOUR WATER
At 1446 rpm
Deflection shown by the vibration meter = 1.82 mm
Dial indicator reading = 1.8 mm
At 847 rpm
Deflection shown by the vibration meter = 1.20 mm
Dial indicator reading = 1.3 mm
At 331 rpm
Deflection shown by the vibration meter = 0.67 mm
Dial indicator reading = 0.6 mm
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21. RESULT AND DISCUSSIONS
IN AIR WITHOUT CRACK
2
1.8
1.6
1.4
Displacement
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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22. IN WATER(WITHOUT CRACK)
2
1.8
1.6
1.4
1.2
Displacement
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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23. IN FLOUR-WATER WITHOUT
CRACK
2
1.8
1.6
1.4
1.2
Displacement
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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24. IN AIR(WITH CRACK)
2
1.8
1.6
1.4
1.2
Displacement
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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25. IN WATER(WITH CRACK)
2
1.8
1.6
1.4
1.2
Displacement
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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26. IN FLOUR-WATER(WITH CRACK)
2
1.8
1.6
1.4
1.2
Displacement
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25
Frequency
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27. CONCLUSION
1)Presence of crack in rotor makes significant difference in
amplitude of vibration to that of uncracked one when rotates in a
fluid medium.
viscosity of fluid medium increases, the critical speed of the rotor
decreases along with the amplitude of vibration.
2)Amplitude of transverse vibration of the rotor system increases
with the increase in the radius of the container carrying the fluid.
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28. CONCLUSION(CONTD.)
3)Due to the presence of crack, the critical speed of rotor decreases.
4)Due to low critical speed the damping coefficient increases for which, the
dimensionless amplitude of the rotating cracked shaft. Due to low critical
speed the damping coefficient increases for which, the dimensionless
amplitude of the rotating cracked shaft is lowest when measured along the
crack direction and is the highest in uncracked one for the same type of
viscous fluid.
5)External damping has got more impact in reducing the amplitude of
vibration than in changing the resonance speed.
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29. FURTHER WORK
1)Bearing characteristics for rotor systems play an important role on its
dynamic behavior, which can be incorporated in the theory for higher
accuracy.
2)Gyroscopic effect which has not been considered in the present analysis can
be taken into account.
3)Stability analysis of cracked structures can be included in the present study.
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30. REFERENCE
KITO, F., Trans Japan Society of Mechanical Engineering (in
Japanese), vol. 22, No. (1956-9), pp663.
Iida, S., Trans, Japan Society of Mechanical Engineers (in
Japanese), Vol.24, No.141 (1958-5), pp278, 283; Vol.25, No.151 (1959-
3), pp.235.
Fritz, R.J., the effects of an annular fluid on the vibrations of a long
rotor, part1-theory, journal of Basic Engineering, Vol.92, No.4 (1970-
12), pp923-929.
Fritz, R.J., the effects of an annular fluid on the vibrations of a long
rotor, part2-test, journal of Basic Engineering, Vol.92, No.4 (1970-12), pp930-
935.
Brennen, C., on the flow in an annulus surrounding a whirling cylinder.
Journal of Fluid Mechanics, Vol.75, part 1, 1976, pp.173-191.
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Walson, W.H., O L O F M E C HandI Clark.L.G.,EdynamicKstabilityVofR rotating shafts
S C H O Ames,W.F. A N C A L E N G I N E R I N G , IIT UNI E SITY

31. REFERENCE(CONTD.)
Crighton,D.G.,Resonant oscillations of fluid-loaded struts, journal of sound
and vibration, vol.87, no.3,1983,pp.429-437.
Achenbach, J.D. and Qu, J., Resonant vibration of a submerged beam, journal
of sound and vibration, vol.105 (2), 1986, pp.185-198.
Shimogo,T. and Krazao,Y., critical speed of rotor in a liquid, Bulletin of the
JSME , Vol.25,No.200,1982,pp 276-283.
Kadyrov S.G., Wauer, J and Sorokin, S.V., a potential technique in the theory
of interaction between a structure and a viscous, compressible fluid, Archive
of Applied Mechanics 71, 2001, pp.405-417.
Seeman, R. and Wauer, J., Finite oscillatory motion of a body immersed in an
inviscid fluid at rest, and Stochastic Dynamics ,AMD-Vol.192/DE-
Vol.78,ASME 1994,pp.135-141.
Seeman, R. and Wauer, J., Fluid- structural coupling of vibrating bodies in
contact with a fluid, Proceeding 3rd Polish-German Workshop on Dynamical
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Problems in Mechanical Systems,1993,pp.31-42.
SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY

32. THANK YOU
We have been highly obliged to undertake this project as
part of our curriculum for B.Tech and would like to
extend our warmest regards for our Mentor-cum-Guide
Prof.Isham Panigrahi and our Respected
Dean.Prof(Dr.)K.C.Singh for helping us through the
entire duration of the project with their expertise and
valuable insights.
Group-14
Mechanical Engg
Batch:2008-2012
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SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY