0
Research Papers: Applications

A Fractional Calculus Model of Viscoelastic Stator Supports Coupled With Elastic Rotor–Stator Rub

[+] Author and Article Information
Patrick A. Smyth

Georgia Institute of Technology,
School of Mechanical Engineering,
Atlanta, GA 30332
e-mail: pasmyth4@gatech.edu

Philip A. Varney

Georgia Institute of Technology,
School of Mechanical Engineering,
Atlanta, GA 30332
e-mail: pvarney3@gatech.edu

Itzhak Green

Georgia Institute of Technology,
School of Mechanical Engineering,
Atlanta, GA 30332
e-mail: itzhak.green@me.gatech.edu

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received June 8, 2015; final manuscript received November 7, 2015; published online June 21, 2016. Assoc. Editor: Bugra Ertas.

J. Tribol 138(4), 041101 (Jun 21, 2016) (8 pages) Paper No: TRIB-15-1187; doi: 10.1115/1.4032787 History: Received June 08, 2015; Revised November 07, 2015

Rotating machinery is inherently susceptible to costly and dangerous faults. One such commonly encountered fault is undesirable dynamic contact between the rotor and stator (i.e., rotor–stator rub). The forces generated during rotor–stator rub are fundamentally tribological, as they are generated by contact and friction and result in wear. These forces are typically found by assuming linear elastic contact and dry Coulomb friction at the rotor–stator interface, where the normal force is a linear function of the interference. For the first time, this work incorporates viscoelasticity into the stator support and investigates its influence on the global dynamics of rotor–stator rub. The viscoelastic stator supports are modeled using fractional calculus, an approach which adeptly and robustly characterizes the viscoelasticity. Specifically, a fractional derivative order of one-half is employed to generate an analytic time-domain form of viscoelastic impedance. This approach directly assimilates viscoelasticity into the system dynamics, since the rotor equations of motion are integrated numerically in the time-domain. The coupled rotor–stator dynamic model incorporating viscoelastic supports is solved numerically to explore the influence of viscoelasticity. This model provides a framework for analysis of dynamic systems where viscoelasticity is included.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Lee, A. S. , and Green, I. , 1994, “ Higher Harmonic Oscillations in a Non-Contacting FMR Mechanical Face Seal Test Rig,” ASME J. Vib. Acoust., 116(2), pp. 161–167. [CrossRef]
Bernardo, M. , Budd, C. , Champneys, A. R. , and Kowalczyk, P. , 2008, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer Science and Business Media, Springer-Verlag, London.
Varney, P. , and Green, I. , 2014, “ Rotor/Stator Rubbing Contact in an Overhung Rotordynamic System,” STLE Annual Meeting, Orlando, FL.
Varney, P. , and Green, I. , 2014, “ Nonlinear Phenomena, Bifurcations, and Routes to Chaos in an Asymmetrically Supported Rotor-Stator Contact System,” J. Sound Vib., 336, pp. 207–226. [CrossRef]
Chu, F. , and Zhang, Z. , 1997, “ Bifurcation and Chaos in a Rub-Impact Jeffcott Rotor System,” J. Sound Vib., 210(1), pp. 1–18. [CrossRef]
Zhang, W. M. , and Meng, G. , 2006, “ Stability, Bifurcation and Chaos of a High-Speed Rub-Impact Rotor System in Mems,” Sens. Actuators, 127(1), pp. 163–178. [CrossRef]
Groll, G. V. , and Ewins, D. J. , 2002, “ A Mechanism of Low Subharmonic Response in Rotor/Stator Contact Measurements and Simulation,” ASME J. Vib. Acoust., 124(3), pp. 350–358. [CrossRef]
Jacquet-Richardet, G. , Torkhani, M. , Cartraud, P. , Thouverez, F. , Baranger, T. N. , Herran, M. , Gibert, C. , Baguet, S. , Almeida, P. , and Peletan, L. , 2013, “ Rotor to Stator Contacts in Turbomachines Review and Application,” Mech. Syst. Signal Process., 40(2), pp. 401–420. [CrossRef]
Yu, J. J. , Goldman, P. , Bently, D. E. , and Muzynska, A. , 2002, “ Rotor/Seal Experimental and Analytical Study on Full Annular Rub,” ASME J. Eng. Gas Turbines Power, 124(2), pp. 340–350. [CrossRef]
Childs, D. W. , and Bhattacharya, A. , 2007, “ Prediction of Dry-Friction Whirl and Whip Between a Rotor and a Stator,” ASME J. Vib. Acoust., 129(3), pp. 355–362. [CrossRef]
Yu, J. , 2012, “ On Occurrence of Reverse Full Annular Rub,” ASME J. Eng. Gas Turbines Power, 134(1), pp. 219–227. [CrossRef]
Cao, J. , Ma, C. , Jiang, Z. , and Liu, S. , 2011, “ Nonlinear Dynamic Analysis of Fractional Order Rub-Impact Rotor System,” Commun. Nonlinear Sci. Numer. Simul., 16(3), pp. 1443–1463. [CrossRef]
Patel, T. H. , Zuo, M. J. , and Zhao, X. , 2012, “ Nonlinear Lateral-Torsional Coupled Motion of a Rotor Contacting a Viscoelastically Suspended Stator,” Nonlinear Dyn., 69(1), pp. 325–339. [CrossRef]
Dutt, J. K. , and Nakra, B. C. , 1992, “ Stability of Rotor Systems With Viscoelastic Supports,” J. Sound Vib., 153(1), pp. 89–96. [CrossRef]
Lee, Y. B. , Kim, T. H. , Lee, N. S. , and Choi, D. H. , 2004, “ Dynamic Characteristics of a Flexible Rotor System Supported by a Viscoelastic Foil Bearing (VEFB),” Tribol. Int., 37(9), pp. 679–687. [CrossRef]
Shabaneh, N. H. , and Zu, J. W. , 2000, “ Dynamic Analysis of Rotor-Shaft Systems With Viscoelastically Supported Bearings,” Mech. Mach. Theory, 35(9), pp. 1313–1330. [CrossRef]
Wilkes, J. , Moore, J. , Ransom, D. , and Vannini, G. , 2014, “ An Improved Catcher Bearing Model and an Explanation of the Forward Whirl/Whip Phenomenon Observed in Active Magnetic Bearing Transient Drop Experiments,” ASME J. Eng. Gas Turbines Power, 136(4), pp. 1–11.
Sun, G. , Palazzolo, A. , Provenza, A. , and Montague, G. , 2004, “ Detailed Ball Bearing Model for Magnetic Suspension Auxiliary Service,” J. Sound Vib., 269(3–5), pp. 933–963.
Beatty, R. F. , 1985, “ Differentiating Rotor Response Due to Radial Rubbing,” J. Vib., Acoust., Stress, Reliab. Des., 107(2), pp. 151–160. [CrossRef]
Smyth, P. A. , Green, I. , Jackson, R. L. , and Hanson, R. R. , 2014, “ Biomimetic Model of Articular Cartilage Based on In Vitro Experiments,” J. Biomimetics, Biomater. Biomed. Eng., 21, pp. 75–91. [CrossRef]
Bagley, R. L. , and Torvik, P. J. , 1979, “ A Generalized Derivative Model for an Elastomer Damper,” Shock Vib. Bull., 49(2), pp. 135–143.
Bagley, R. L. , and Torvik, P. J. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol. (1978-Present), 27(3), pp. 201–210. [CrossRef]
Rogers, L. , 1983, “ Operators and Fractional Derivatives for Viscoelastic Constitutive Equations,” J. Rheol., 27(4), pp. 351–372. [CrossRef]
Koeller, R. , 1984, “ Applications of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51(2), pp. 299–307. [CrossRef]
Torvik, P. J. , and Bagley, R. L. , 1984, “ On the Appearance of the Fractional Derivative in the Behavior of Real Materials,” ASME J. Appl. Mech., 51(2), pp. 294–298. [CrossRef]
Bagley, R. L. , and Torvik, P. J. , 1985, “ Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA J., 23(6), pp. 918–925. [CrossRef]
Bagley, R. L. , and Torvik, P. J. , 1986, “ On the Fractional Calculus Model of Viscoelastic Behavior,” J. Rheol. (1978-Present), 30(1), pp. 133–155. [CrossRef]
Koeller, R. C. , 1986, “ Polynomial Operators, Stieltjes Convolution, and Fractional Calculus in Hereditary Mechanics,” Acta Mech., 58(3–4), pp. 251–264. [CrossRef]
Bagley, R. L. , 1989, “ Power Law and Fractional Calculus Model of Viscoelasticity,” AIAA J., 27(10), pp. 1412–1417. [CrossRef]
Erdelyi, A. , Magnus, W. , Oberhettinger, F. , and Tricomi, F. , eds., 1955, Higher Transcendental Functions, Vol. III, McGraw-Hill, New York.
Podlubny, I. , 1998, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Academic Press, San Diego, CA.
Szumski, R. G. , and Green, I. , 1991, “ Constitutive Laws in Time and Frequency Domains for Linear Viscoelastic Materials,” J. Acoust. Soc. Am., 90(40), p. 2292. [CrossRef]
Szumski, R. G. , 1993, “ A Finite Element Formulation for the Time Domain Vibration Analysis of an Elastic-Viscoelastic Structure,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA.
Biesel, V. , 1993, “ Experimental Measurement of the Dynamic Properties of Viscoelastic Materials,” M.S. thesis, Georgia Institute of Technology, Atlanta, GA.
Smyth, P. A. , and Green, I. , 2015, “ A Fractional Calculus Model of Articular Cartilage Based on Experimental Stress-Relaxation,” Mech. Time-Depend. Mater., 19(2), pp. 209–228. [CrossRef]
Scholz, A. , 2011, “ Ein beitrag zur optimierung des schwingungsverhaltens komplexer rotorsysteme mit viskoelastischen dämpfungselementen,” Ph.D. thesis, Technische Universitat Berlin, Berlin.
Liebich, R. , Scholz, A. , and Wieschalla, M. , 2012, “ Rotors Supported by Elastomer-Ring-Dampers: Experimental and Numerical Investigations,” 10th International Conference on Vibrations in Rotating Machinery, London, pp. 443–453.
Pooseh, S. , Almeida, R. , and Torres, D. F. M. , 2013, “ Numerical Approximations of Fractional Derivatives With Applications,” Asian J. Control, 15(3), pp. 698–712. [CrossRef]
Popprath, S. , and Ecker, H. , 2007, “ Nonlinear Dynamics of a Rotor Contacting an Elastically Suspended Stator,” J. Sound Vib., 308(3–5), pp. 767–784. [CrossRef]
Chu, F. , and Zhang, Z. , 1997, “ Periodic, Quasi-Periodic and Chaotic Vibrations of a Rub-Impact Rotor System Supported on Oil Film Bearings,” Int. J. Eng. Sci., 35(9), pp. 963–973. [CrossRef]
Goldman, P. , and Muszynska, A. , 1994, “ Dynamic Effects in Mechanical Structures With Gaps and Impacting: Order and Chaos,” ASME J. Vib. Acoust., 116(4), pp. 541–547. [CrossRef]
Chang-Jian, C. W. , and Chen, C. K. , 2007, “ Chaos and Bifurcation of a Flexible Rub-Impact Rotor Supported by Oil Film Bearings With Nonlinear Suspension,” Mech. Mach. Theory, 42(3), pp. 312–333. [CrossRef]
Kim, Y. B. , and Noah, S. T. , 1990, “ Bifurcation Analysis for a Modified Jeffcott Rotor With Bearing Clearances,” Nonlinear Dyn., 1(3), pp. 221–241. [CrossRef]
Inayat-Hussain, J. I. , 2010, “ Bifurcations in the Response of a Jeffcott Rotor With Rotor-to-Stator Rub,” ASME Paper No. ESDA2010-24453.
Chavez, J. P. , and Wiercigroch, M. , 2013, “ Bifurcation Analysis of Periodic Orbits of a Non-Smooth Jeffcott Rotor Model,” Commun. Nonlinear Sci. Numer. Simul., 18(9), pp. 2571–2580. [CrossRef]
Abu-Mahfouz, I. , and Banerjee, A. , 2013, “ On the Investigation of Nonlinear Dynamics of a Rotor With Rub-Impact Using Numerical Analysis and Evolutionary Algorithms,” Proc. Comput. Sci., 20, pp. 140–147. [CrossRef]
Chang-Jian, C. W. , and Chen, C. K. , 2009, “ Chaos of Rub-Impact Rotor Supported by Bearings With Nonlinear Suspension,” Tribol. Int., 42(3), pp. 426–439. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Jeffcott rotor with corresponding viscoelastically suspended stator

Grahic Jump Location
Fig. 2

Viscoelastically suspended stator

Grahic Jump Location
Fig. 3

Rotor–stator parameters (for clarity here, the viscoelastic stator supports are shown in Fig. 2)

Grahic Jump Location
Fig. 4

Vector diagram showing the relationship between rotor and stator deflections

Grahic Jump Location
Fig. 5

Mechanical representation of viscoelastic models (a) SLS (prony series, n = 1) and (b) fractional model (n = 1)

Grahic Jump Location
Fig. 6

Comparing the rigidly and flexibly mounted stators using parameters provided in Table 1 (responses calculated at n/ωn = 1.1). (a) Rigidly mounted stator, where the dashed circle represents the clearance and (b) flexibly mounted stator (for validation purposes, compare the response shown here to those provided by Popprath and Ecker [39]).

Grahic Jump Location
Fig. 7

Example periodic response for the viscoelastically suspended stator system (n/ωn = 1.6): (a) rotor and stator orbits and (b) rotor frequency spectrum

Grahic Jump Location
Fig. 8

Example quasiperiodic response for the viscoelastically suspended stator system (n/ωn = 1.5): (a) rotor and stator orbits and (b) rotor frequency spectrum

Grahic Jump Location
Fig. 9

Example chaotic response for the viscoelastically suspended stator system (n/ωn = 1.25): (a) rotor and stator orbits and (b) rotor frequency spectrum

Grahic Jump Location
Fig. 10

Shaft speed bifurcation diagram showing numerous bifurcations and a generally rich nonlinear response

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In