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A shaft finite element for analysis of viscoelastic tapered Timoshenko rotors

  • Amit BhowmickEmail author
  • Smitadhi Ganguly
  • Sumanta Neogy
  • Arghya Nandi
Technical Paper
  • 54 Downloads

Abstract

Finite element modeling of thick rotor (deep rotor) requires the use of the Timoshenko element, which takes care of shear deformation. Modeling a rotor whose cross section varies can be done using uniform stepped cylindrical elements, but the number of elements required for the converged results is enormous. Further modeling of doubly tapered rotors and rotors with sharp changes in taper angle using such methods become difficult and does not appear feasible that is why the formulation of conical rotor elements is necessary. The complexity in formulations arises due to the presence of the shear parameter in the denominator. Till now, in the literature of the Timoshenko rotor, geometric properties are taken to be linearly varying; the shear parameter is the approximation of average values of two end nodes. On the other hand, the modeling of damping in rotors becomes essential as it controls the stability limit. The present paper develops three different viscoelastic tapered rotor elements using the Maxwell–Wiechert model. While the first element (VTRE 1) uses the average shear parameter, the second element (VTRE 2) uses a higher degree polynomial to approximate the shear parameter. While the first two-rotor elements are solid, the third element (VTRE 3) is hollow. Results show that viscoelastic tapered elements provide better results with the lesser number of elements. For the rotor with a large taper angle, VTRE 2 performs better than VTRE 1. Further modeling of the hollow tapered rotor with a linear variation of taper angle and discontinuities at definite points is not possible by the use of uniform elements. Modeling such a rotor using VTRE 3 is rendered simple. Practical usage of such rotors further enhances the importance of the present work.

Keywords

Maxwell–Wiechert model Viscoelastic tapered rotor element Timoshenko beam theory 

Notes

References

  1. 1.
    Williams ML (1964) Structural analysis of viscoelastic materials. AIAA J 2(5):785–808CrossRefGoogle Scholar
  2. 2.
    Golla DF, Hughes PC (1985) Dynamics of viscoelastic structures—a time-domain, finite element formulation. J Appl Mech 52(4):897–906MathSciNetCrossRefGoogle Scholar
  3. 3.
    McTavish DJ, Hughes PC (1993) Modeling of linear viscoelastic space structures. J Vib Acoust 115(1):103–110CrossRefGoogle Scholar
  4. 4.
    Adhikari S (2001) Eigenrelations for nonviscously damped systems. AIAA J 39(8):1624–1630CrossRefGoogle Scholar
  5. 5.
    Adhikari S (2002) Dynamics of nonviscously damped linear systems. J Eng Mech 128(3):328–339CrossRefGoogle Scholar
  6. 6.
    Wagner N, Adhikari S (2003) Symmetric state-space method for a class of nonviscously damped systems. AIAA J 41(5):951–956CrossRefGoogle Scholar
  7. 7.
    Adhikari S, Wagner N (2003) Analysis of asymmetric nonviscously damped linear dynamic systems. J Appl Mech 70(6):885–893CrossRefGoogle Scholar
  8. 8.
    Adhikari S (2005) Qualitative dynamic characteristics of a non-viscously damped oscillator. Proc R Soc A Math Phys Eng Sci 461(2059):2269–2288MathSciNetCrossRefGoogle Scholar
  9. 9.
    Adhikari S (2008) Dynamic response characteristics of a nonviscously damped oscillator. J Appl Mech 75(1):11003CrossRefGoogle Scholar
  10. 10.
    De Lima A, Stoppa M, Rade D, Steffen V Jr (2006) Sensitivity analysis of viscoelastic structures. Shock Vib 13(4–5):545–558CrossRefGoogle Scholar
  11. 11.
    Lesieutre GA, Mingori DL (1990) Finite element modeling of frequency-dependent material damping using augmenting thermodynamic fields. J Guid Control Dyn 13(6):1040–1050CrossRefGoogle Scholar
  12. 12.
    Lesieutre GA, Bianchini E (1995) Time domain modeling of linear viscoelasticity using anelastic displacement fields. J Vib Acoust 117:424–430CrossRefGoogle Scholar
  13. 13.
    Lesieutre GA, Bianchini E, Maiani A (1996) Finite element modeling of one-dimensional viscoelastic structures using anelastic displacement fields. J Guid Control Dyn 19(3):520–527CrossRefGoogle Scholar
  14. 14.
    Rusovici R (1999) Modeling of shock wave propagation and attenuation in viscoelastic structures. Ph.D. thesis, Virginia TechGoogle Scholar
  15. 15.
    Roy H, Dutt JK, Datta PK (2009) Dynamics of multilayered viscoelastic beams. Struct Eng Mech 33(4):391–406CrossRefGoogle Scholar
  16. 16.
    Friswell MI, Dutt JK, Adhikari S, Lees AW (2010) Time domain analysis of a viscoelastic rotor using internal variable models. Int J Mech Sci 52(10):1319–1324CrossRefGoogle Scholar
  17. 17.
    Kliem W (1987) The dynamics of viscoelastic rotors. Dyn Stab Syst 2(2):424–429zbMATHGoogle Scholar
  18. 18.
    Sinha SK (1989) Stability of a viscoelastic rotor-disk system under dynamic axial loads. AIAA J 27(11):1653–1655CrossRefGoogle Scholar
  19. 19.
    Sturla F, Argento A (1996) Free and forced vibrations of a spinning viscoelastic beam. J Vib Acoust 118(3):463–468CrossRefGoogle Scholar
  20. 20.
    Shabaneh N, Zu JW (2000) Dynamic and stability analysis of rotor-shaft systems with viscoelastically supported bearings. Trans Can Soc Mech Eng 24(1B):179–189CrossRefGoogle Scholar
  21. 21.
    Shabaneh N, Zu JW (2000) Dynamic analysis of rotor-shaft systems with viscoelastically supported bearings. Mech Mach Theory 35(9):1313–1330CrossRefGoogle Scholar
  22. 22.
    Bavastri CA, Ferreira EMDS, Espíndola JJD, Lopes EMDO (2008) Modeling of dynamic rotors with flexible bearings due to the use of viscoelastic materials. J Braz Soc Mech Sci Eng 30(1):22–29CrossRefGoogle Scholar
  23. 23.
    Genta G, Delprete C, Bassani D (1996) Dynrot: a finite element code for rotordynamic analysis based on complex co-ordinates. Eng Comput 13(6):86–109CrossRefGoogle Scholar
  24. 24.
    Genta G, Amati N (2010) Hysteretic damping in rotordynamics: an equivalent formulation. J Sound Vib 329(22):4772–4784CrossRefGoogle Scholar
  25. 25.
    Kang C, Hsu W, Lee E, Shiau T (2011) Dynamic analysis of gear-rotor system with viscoelastic supports under residual shaft bow effect. Mech Mach Theory 46(3):264–275CrossRefGoogle Scholar
  26. 26.
    Roy H, Dutt JK, Chandraker S (2012) Modelling of multilayered viscoelastic rotors – an operator based approach. In: 8th international conference on vibration engineering and technology of machinery (VETOMACVIII), Gdansk, Poland, 3–6 Sept 2012Google Scholar
  27. 27.
    Ganguly S, Nandi A, Neogy S (2016) A state space viscoelastic shaft finite element for analysis of rotors. Proced Eng 144:374–381CrossRefGoogle Scholar
  28. 28.
    Bhowmick A, Nandi A, Neogy S, Ganguly S (2018) A shaft finite element for analysis of viscoelastic tapered and hollow tapered rotors. In: International conference on mechanical engineering, pp 457–482, SpringerGoogle Scholar
  29. 29.
    Nelson H (1980) A finite rotating shaft element using timoshenko beam theory. J Mech Des 102(4):793–803Google Scholar
  30. 30.
    Lalanne M, Ferraris G (1998) Rotordynamics prediction in engineering, vol 2. Wiley, HobokenGoogle Scholar
  31. 31.
    Hong S-W, Park J-H (1999) Dynamic analysis of multi-stepped, distributed parameter rotor-bearing systems. J Sound Vib 227(4):769–785CrossRefGoogle Scholar
  32. 32.
    Xiong G, Yi J, Zeng C, Guo H, Li L (2003) Study of the gyroscopic effect of the spindle on the stability characteristics of the milling system. J Mater Process Technol 138(1–3):379–384CrossRefGoogle Scholar
  33. 33.
    Cavalini AA Jr, Galavotti TV, Morais TS, Koroishi EH, Steffen V Jr (2011) Vibration attenuation in rotating machines using smart spring mechanism. Math Probl Eng.  https://doi.org/10.1155/2011/340235 CrossRefzbMATHGoogle Scholar
  34. 34.
    Cavalini AA Jr, Lobato FS, Koroishi EH, Steffen V Jr (2016) Model updating of a rotating machine using the self-adaptive differential evolution algorithm. Inverse Probl Sci Eng 24(3):504–523CrossRefGoogle Scholar
  35. 35.
    Morais TS, Steffen V Jr, Mahfoud J (2012) Control of the breathing mechanism of a cracked rotor by using electro-magnetic actuator: numerical study. Latin Am J Solids Struct 9(5):581–596CrossRefGoogle Scholar
  36. 36.
    Dakel M, Baguet S, Dufour R (2014) Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings. J Sound Vib 333(10):2774–2799CrossRefGoogle Scholar
  37. 37.
    Lara-Molina FA, Koroishi EH, Steffen V Jr (2015) Uncertainty analysis of flexible rotors considering fuzzy parameters and fuzzy-random parameters. Latin Am J Solids Struct 12(10):1807–1823CrossRefGoogle Scholar
  38. 38.
    Ganguly S, Nandi A, Neogy S (2018) A viscoelastic timoshenko shaft finite element for dynamic analysis of rotors. J Vib Control 24(11):2180–2200MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rouch K, Kao J-S (1979) A tapered beam finite element for rotor dynamics analysis. J Sound Vib 66(1):119–140CrossRefGoogle Scholar
  40. 40.
    Edney S, Fox C, Williams E (1990) Tapered timoshenko finite elements for rotor dynamics analysis. J Sound Vib 137(3):463–481CrossRefGoogle Scholar
  41. 41.
    Bazoune A, Khulief Y, Stephen N (2003) Shape functions of three-dimensional timoshenko beam element. J Sound Vib 259(2):473–480CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.Department of Mechanical EngineeringHooghly Engineering and Technology CollegeHooghlyIndia
  3. 3.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

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