# Dynamic study of viscoelastic rotor: a comparative study using analytical and finite element model considering higher-order system

## Abstract

In the past, many researchers developed rotor models using either lump system or finite element approach, where material damping played a crucial role in dynamic behaviour. Such damping in any rotating structure triggers instability at the supercritical range. In most of the literatures, material damping has been incorporated either by frequency-independent hysteretic damping or frequency-dependent viscous damping, but these models are insufficient to estimate the dynamic characteristics of the system. The motivation for using general viscoelastic model arises from a need to capture the influence of both types of damping. Such type of modelling is done through operator-based constitutive relationship. The numerator and denominator of material modulus are a polynomial of differential time operator, and polynomial coefficients are known as a viscoelastic parameter. The operator-based constitutive relationship is further utilized to bring down higher-order equations of motion by using two different techniques, i.e. (a) analytical approach and (b) finite element approach.The shaft damping is tackled in such a manner that the dissipation effects can be considered through all coordinates. The significance of both approaches is explained with the help of stability and response analysis at various disc positions.

## Keywords

Analytical model Finite element model Viscoelastic rotor Effective mass Effective diametral mass moment of inertia Stability analysis## References

- 1.Asnani, N.T.: Vibration analysis of multilayered beams with constrained viscoelastic layers. Ph.D. Thesis, Indian Institute of Technology, Delhi (1971)Google Scholar
- 2.Bland, D.R.: The Theory of Linear Viscoelasticity, pp. 1–10. Pergamon Press, Oxford (1960)zbMATHGoogle Scholar
- 3.Chandraker, S.: Modelling and Model Reduction of Viscoelastic Composite Rotors: An Operator-Based Approach. Ph.D. Thesis from National Institute of Technology, Rourkela, July (2016)Google Scholar
- 4.Chandraker, S., Roy, H., Maurya, G.: Modal analysis of multi-layer viscoelastic rotors considering higher order model. In: Proceeding in GTINDIA, ASME Conference, Bangalore, India (2013)Google Scholar
- 5.Combescure, D., Lazarus, A.: Refined finite element modelling for the vibration analysis of large rotating machines: application to the gas turbine modular helium reactor power conversion unit. J. Sound Vib.
**318**, 1262–1280 (2008)CrossRefGoogle Scholar - 6.Dutt, J.K., Nakra, B.C.: Stability of rotor systems with viscoelastic supports. J. Sound Vib.
**153**, 89–96 (1992)CrossRefzbMATHGoogle Scholar - 7.Dutt, J.K., Nakra, B.C.: Dynamics of rotor shaft-system on flexible support with gyroscopic effects. Mech. Res. Commun.
**22**, 541–545 (1995)CrossRefGoogle Scholar - 8.Dutt, J.K., Roy, H.: Viscoelastic modelling of rotor-shaft systems using an operator-based approach. Proc. Inst. Mech. Eng. Part C
**225**, 73–87 (2011)CrossRefGoogle Scholar - 9.Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Rotor Dynamics: Modeling and Analysis of Rotating Machines. Cambridge University Press, Cambridge (2010a)CrossRefzbMATHGoogle Scholar
- 10.Friswell, M.I., Dutt, J.K., Adhikari, S., Lees, A.W.: Time domain analysis of a viscoelastic rotor using internal variable models. Int. J. Mech. Sci.
**52**, 1319–1324 (2010b)CrossRefGoogle Scholar - 11.Genta, G.: On a persistent misunderstanding of the role of hysteretic damping in rotor dynamics. J. Vib. Acoust.
**126**, 459–461 (2004)CrossRefGoogle Scholar - 12.Genta, G.: Dynamics of Rotating Systems. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
- 13.Genta, G., Amati, N.: On the equivalent viscous damping for the system with hysteretic. In: Meccanica Dei Solidi, Atti dell’Accademia delle Scienze di Torino, pp. 21–40. (2008)Google Scholar
- 14.Genta, G., Amati, N.: Hysteretic damping in rotordynamics: an equivalent formulation. J. Sound Vib.
**329**, 4772–4784 (2010)CrossRefGoogle Scholar - 15.Grybos, R.: The dynamics of a viscoelastic rotor in flexible bearing. Arch. Appl. Mech.
**61**, 479–487 (1991)Google Scholar - 16.Kapur, A.D., Nakra, B.C., Chawla, D.R.: Shock response of viscoelastically damped beam. Journal of Sound and Vibration
**55**(3), 351–362 (1977)CrossRefzbMATHGoogle Scholar - 17.Ku, D.M.: Finite element analysis of whirl speeds for rotor-bearing systems with internal damping. Mech. Syst. Signal Process.
**12**, 599–610 (1998)CrossRefGoogle Scholar - 18.Kramer, E.: Dynamics of Rotors and Foundation. Springer, Berlin (1993)CrossRefGoogle Scholar
- 19.Lalanne, M., Ferraris, G.: Rotor Dynamics Prediction in Engineering. Wiley, Hoboken (1998)Google Scholar
- 20.Lesieutre, G.A., Mingori, D.L.: Finite element modelling of frequency-dependent material damping using augmenting thermodynamic fields. J. Guid. Control Dyn.
**13**, 1040–1050 (1990)CrossRefzbMATHGoogle Scholar - 21.Lesieutre, G.A., Bianchini, E., Maiani, A.: Finite element modelling of one-dimensional viscoelastic structures using anelastic displacement fields. J. Guid. Control Dyn.
**19**, 520–527 (1996)CrossRefzbMATHGoogle Scholar - 22.Meirovitch, L.: Elements of Vibration Analysis, International edn. McGraw-Hill, New York (1986)zbMATHGoogle Scholar
- 23.Ozguven, H.N., Ozkan, Z.L.: Whirl speeds and unbalance response of multibearing rotors using finite elements. J. Vib. Acoust.
**106**, 72–79 (1984)CrossRefGoogle Scholar - 24.Roy, H.: Study of Dynamics of Viscoelastic Rotors: A Finite Element Approach. Ph.D. Thesis from Indian Institute of Technology, Kharagpur, India (2008)Google Scholar
- 25.Roy, H., Dutt, J.K., Datta, P.K.: Dynamics of a viscoelastic rotor shaft using augmenting thermodynamic fields—a finite element approach. Int. J. Mech. Sci.
**50**, 845–853 (2008)CrossRefzbMATHGoogle Scholar - 26.Roy, H., Chandraker, S., Dutt, J.K., Roy, T.: Dynamics of multilayer, multidisc viscoelastic rotor—an operator based higher order classical model. J. Sound Vib.
**369**, 87–108 (2016)CrossRefGoogle Scholar - 27.Roy, H., Dutt, J.K., Chandraker, S.: Modelling of multilayered viscoelastic rotors—an operator-based approach. J. Vib. Eng. Technol.
**2**, 485–494 (2014)Google Scholar - 28.Roy, H., Chandraker, S.: A comparative study between classical and finite element model for multilayer viscoelastic rotors. In: Published Proceeding at ASME GTINDIA Conference, Hyderabad, India, paper id-GTINDIA-1330 (2015)Google Scholar
- 29.Roy, H., Dutt, J.K.: Dynamics of polymer and polymer composite rotors—an operator based finite element approach. Appl. Math. Model.
**40**, 1754–1768 (2016)MathSciNetCrossRefGoogle Scholar - 30.Roy, H., Chandraker, S.: Dynamic study of viscoelastic rotor: modal analysis of higher order model considering various asymmetries. Mech. Mach. Theory
**109**, 65–77 (2017)CrossRefGoogle Scholar - 31.Shames, I.H., Cozzarelli, F.A.: Elastic and Inelastic Stress Analysis. Prentice Hall, Englewood Cliffs (1992)zbMATHGoogle Scholar
- 32.Srinath, L.S.: Advanced Mechanics of Solids. Tata McGraw Hill, New Delhi (2008)Google Scholar
- 33.Tondl, A.: Some Problem of Rotor Dynamics. Chapman and Hall, London (1965)Google Scholar
- 34.Zhou, X.Q., Yu, D.Y., Shao, X.Y., Zhang, S.Q., Wang, S.: Research and applications of viscoelastic vibration damping materials: a review. Compos. Struct.
**136**, 460–480 (2016)CrossRefGoogle Scholar - 35.Zorzi, E.S., Nelson, H.D.: Finite element simulation of rotor-bearing systems with internal damping. J. Eng. Gas Turbine Power
**99**, 71–76 (1977)CrossRefGoogle Scholar