B.tech(Vibrational Analysis)-2012

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

Under the expert guidance and mentorship of

Prof. Isham Panigrahi

VIBRATIONAL ANALYSIS OF CANTILEVER ROTOR IN VISCOUS MEDIUM

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

INTRODUCTION-CRACKS IN A SHAFT/BEAM

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

PRACTICAL APPLICATIONS OF CRACK INVESTIGATION IN SHAFTS AND

BEAMS

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

OBJECTIVES OF THE PROJECT

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

IMPORTANT THEORIES AND METHODS USED FOR THE ANALYSIS

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

IMPORTANT THEORIES AND METHODS USED FOR THE ANALYSIS

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

An electric motor

(203v,50hz,0.75amps,

1/8hp,93w,1350rpm)

A flexible coupling

APPARATUS REQUIRED

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

Bearing with housing

Dial indicator

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

Stroboscope

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

EXPERIMENTAL SETUP

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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 .

THEORETICAL CALCULATION

APPROACH #1

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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 fn = 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.

APPROACH #1(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

Now following another approach to find out the natural frequency

Stiffness of any beam is given by Kt = 3.14*G*d4 / 32*l

Kt = 3140

Polar moment of inertia of a beam is given by J0 = 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 fn = Wn / 2*3.14

fn= 216.411 Hz

Therefore the theoretical rpm at which the first frequency occurs is = fn * 60

= 12984.66 rpm.

APPROACH #2

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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

EXPERIMENTAL ANALYSIS

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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

EXPERIMENT ANALYSIS(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

3. Deflection in flour-water

At 1446 rpm Deflection shown by the vibration meter = 0.57 mm Dial indicator reading = 0.6 mmAt 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

EXPERIMENTAL ANALYSIS(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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 mmAt 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

EXPERIMENTAL ANALYSIS(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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

EXPERIMENTAL ANALYSIS(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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

EXPERIMENTAL ANALYSIS(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

IN AIR WITHOUT CRACK

Frequency

Dis

pla

cem

ent

RESULT AND DISCUSSIONS

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency

Dis

pla

cem

ent

IN WATER(WITHOUT CRACK)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

IN FLOUR-WATER WITHOUT CRACK

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency

Dis

pla

cem

ent

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency

Dis

pla

cem

ent

IN AIR(WITH CRACK)

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IN WATER(WITH CRACK)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency

Dis

pla

cem

ent

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IN FLOUR-WATER(WITH CRACK)

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency

Dis

pla

cem

ent

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

CONCLUSION

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

CONCLUSION(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

FURTHER WORK

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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.

Walson, W.H., Ames,W.F. and Clark.L.G., dynamic stability of rotating shafts in viscous fluids, journal of applied

Mechanics, june 1964,pp.291-299.

REFERENCE

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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 Problems in Mechanical Systems,1993,pp.31-42.

REFERENCE(CONTD.)

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S C H O O L O F M E C H A N I C A L E N G I N E E R I N G , K I I T U N I V E R SI T Y

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-14Mechanical EnggBatch:2008-2012

THANK YOU