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  • 1. VIBRATIONAL ANALYSIS OF CANTILEVER ROTOR IN VISCOUS MEDIUM Under the expert guidance and mentorship of Prof. Isham Panigrahi 1SCHOOL 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. 2 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 3. PRACTICAL APPLICATIONS OF CRACK INVESTIGATION IN SHAFTS AND BEAMSThe 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 bedIII) preventing failure of rotors used in machineries subjected to various environmental conditions. 3 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 4. OBJECTIVES OF THE PROJECTThe 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. 4 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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. 5 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 6. IMPORTANT THEORIES AND METHODS USED FOR THE ANALYSIS The dynamic response of rotors with transverse cracks rotating in viscousmedium, the amplitude of vibration of rotors are found using Navier-Stokes equation and Fourier transform technique. 6 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 7. APPARATUS REQUIRED An electric motor (203v,50hz,0.75amps, 1/8hp,93w,1350rpm) A flexible coupling 7SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 8. Bearing with housing Dial indicator 8SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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 9SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 10. Stroboscope 10SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 11. EXPERIMENTAL SETUP 11SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 12. THEORETICAL CALCULATION APPROACH #1Volume 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 / m3Theoretically, C = (Modulus of rigidity / density of the rotor ) 1/2 = (80*109/7360.2)1/2 = 3.29*103 m/s . 12 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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 HzTherefore the theoretical rpm at which the first frequency occurs is = fn * 60 = 120955.4 rpm. 13 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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 HzTherefore the theoretical rpm at which the first frequency occurs is = f n * 60 14 = 12984.66 rpm. SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 15. EXPERIMENTAL ANALYSISWITHOUT CRACK IN AIR1.Deflection in airAt 1446 rpm Deflection shown by the vibration meter = 0.05mm Dial indicator reading = 0.1 mmAt 847 rpm Deflection shown by the vibration meter = 0.95 mm Dial indicator reading = 0.9 mmAt 331 rpm Deflection shown by the vibration meter = 1.123 mm Dial indicator reading = 1.2 mm 15 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 16. EXPERIMENT ANALYSIS(CONTD.)2 .Deflection in waterAt 1446 rpm Deflection shown by the vibration meter = 0.18 mm Dial indicator reading = 0.2 mmAt 847 rpm Deflection shown by the vibration meter = 0.78mm Dial indicator reading = 0.7 mmAt 331 rpm Deflection shown by the vibration meter = 1.87 mm Dial indicator reading = 1.9 mm 16 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 17. EXPERIMENTAL ANALYSIS(CONTD.)3. Deflection in flour-waterAt 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 mmAt 331 rpm Deflection shown by the vibration meter = 1.93 mm Dial indicator reading = 1.9 mm 17 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 18. EXPERIMENTAL ANALYSIS(CONTD.)WITH CRACK IN ROTOR1. DEFLECTION IN AIRAt 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 mmAt 331 rpm Deflection shown by the vibration meter = 1.341 mm Dial indicator reading = 1.5 mm 18 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 19. EXPERIMENTAL ANALYSIS(CONTD.)2.DEFLECTION IN WATERAt 1446 rpm Deflection shown by the vibration meter = 0.23 mm Dial indicator reading = 0.3 mmAt 847 rpm Deflection shown by the vibration meter = 0.95 mm Dial indicator reading = 0.9 mmAt 331 rpm Deflection shown by the vibration meter = 1.8 mm Dial indicator reading = 1.8 mm 19 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 20. EXPERIMENTAL ANALYSIS(CONTD.)3. DEFLECTION IN FLOUR WATERAt 1446 rpm Deflection shown by the vibration meter = 1.82 mm Dial indicator reading = 1.8 mmAt 847 rpm Deflection shown by the vibration meter = 1.20 mm Dial indicator reading = 1.3 mmAt 331 rpm Deflection shown by the vibration meter = 0.67 mm Dial indicator reading = 0.6 mm 20 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 21. RESULT AND DISCUSSIONS IN AIR WITHOUT CRACK 2 1.8 1.6 1.4Displacement 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 21 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 22. IN WATER(WITHOUT CRACK) 2 1.8 1.6 1.4 1.2Displacement 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 22 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 23. IN FLOUR-WATER WITHOUT CRACK 2 1.8 1.6 1.4 1.2Displacement 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 23 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 24. IN AIR(WITH CRACK) 2 1.8 1.6 1.4 1.2Displacement 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 24 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 25. IN WATER(WITH CRACK) 2 1.8 1.6 1.4 1.2Displacement 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 25 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 26. IN FLOUR-WATER(WITH CRACK) 2 1.8 1.6 1.4 1.2Displacement 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 Frequency 26 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 27. CONCLUSION 1)Presence of crack in rotor makes significant difference inamplitude 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 increaseswith the increase in the radius of the container carrying the fluid. 27 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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 dimensionlessamplitude 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. 28 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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. 29 SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 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 longrotor, 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. 30 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 theoryof 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 incontact with a fluid, Proceeding 3rd Polish-German Workshop on Dynamical 31 Problems in Mechanical Systems,1993,pp.31-42. SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY
  • 32. THANK YOUWe have been highly obliged to undertake this project as part of our curriculum for B.Tech and would like toextend 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 32SCHOOL OF MECHANICAL ENGINEERING, KIIT UNIVERSITY