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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.

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References

Figures

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Fig. 1

Jeffcott rotor with corresponding viscoelastically suspended stator

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Fig. 2

Viscoelastically suspended stator

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Fig. 3

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

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Fig. 4

Vector diagram showing the relationship between rotor and stator deflections

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Fig. 5

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

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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]).

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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

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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

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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

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Fig. 10

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

Tables

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