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Abstract

The increasing demand for high-speed rotor-bearing systems results in the application of complex materials, which allow for a better control of the vibrational characteristics. This paper presents a model of a rotor including viscoelastic materials and valid up to high spin speeds. Regarding the destabilization of rotor-bearing systems, two main effects have to be investigated, which are strongly related to the associated internal and external damping of the rotor. For this reason, the internal material damping is modeled using fractional time derivatives, which can represent a large class of viscoelastic materials over a wide frequency range. In this paper, the Numerical Assembly Technique (NAT) is extended for the rotating viscoelastic Timoshenko beam with fractional derivative damping. An efficient and accurate simulation of the proposed rotor-bearing model is achieved. Several numerical examples are presented and the influence of internal damping on the rotor-bearing system is investigated and compared to classical damping models.
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Authors and Affiliations

Gregor Überwimmer
1
ORCID: ORCID
Georg Quinz
1
Michael Klanner
1
ORCID: ORCID
Katrin Ellermann
1

  1. Graz University of Technology, Institute of Mechanics, Kopernikusgasse 24/IV, 8010 Graz, Austria
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Abstract

In this paper, a new application of the Numerical Assembly Technique is presented for the balancing of linear elastic rotor-bearing systems with a stepped shaft and arbitrarily distributed mass unbalance. The method improves existing balancing techniques by combining the advantages of modal balancing with the fast calculation of an efficient numerical method. The rotating stepped circular shaft is modelled according to the Rayleigh beam theory. The Numerical Assembly Technique is used to calculate the steady-state harmonic response, eigenvalues and the associated mode shapes of the rotor. The displacements of a simulation are compared to measured displacements of the rotor-bearing system to calculate the generalized unbalance for each eigenvalue. The generalized unbalances are modified according to modal theory to calculate orthogonal correction masses. In this manner, a rotor-bearing system is balanced using a single measurement of the displacement at one position on the rotor for every critical speed. Three numerical examples are used to show the accuracy and the balancing success of the proposed method.
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Bibliography

  1.  J. Tessarzik, Flexible rotor balancing by the exact point speed influence coefficient method. Latham: Mechanical Technology Incorporated, 1972.
  2.  P. Gnielka, “Modal balancing of flexible rotors without test runs: An experimental investigation,” Journal of Vibrations, vol. 90, no. 2, pp. 152–170, 1982.
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  6.  A.-C. Lee, Y.-P. Shih, and Y. Kang, “The analysis of linear rotor bearing systems: A general Transfer Matrix Method,” Journal of Vibration and Accoustics, vol. 115, no. 4, pp. 490–497, 1993.
  7.  J.-S. Wu and H. M. Chou, “A new approach for determining the natural frequency of mode shapes of a uniform beam carrying any number of sprung masses,” Journal of Sound and Vibration, vol.  220, no. 3, pp. 451–468, 1999.
  8.  J.-S. Wu, F.-T. Lin, and H.-J. Shaw, “Analytical solution for whirling speeds and mode shapes of a distributed-mass shaft with arbitrary rigid disks,” Journal of Applied Mechanics, vol. 81, no. 3, pp. 034 503–1–034 503–10, 2014.
  9.  M. Klanner, M.S. Prem, and K. Ellermann, “Steady-state harmonic vibrations of a linear rotor- bearing system with a discontinuous shaft and arbitrarily distributed mass unbalance,” in Proceedings of ISMA2020 International Conference on Noise and Vibration Engineering and USD2020 International Conference on Uncertainty in Structural Dynamics, 2020, pp. 1257–1272.
  10.  M. Klanner and K. Ellermann, “Steady-state linear harmonic vibrations of multiple-stepped Euler-Bernoulli beams under arbitrarily distributed loads carrying any number of concentrated elements,” Applied and Computational Mechanics, vol. 14, no. 1, pp. 31–50, 2019.
  11.  M.B. Deepthikumar, A.S. Sekhar, and M.R. Srikanthan, “Modal balancing of flexible rotors with bow and distributed unbalance,” Journal of Sound and Vibration, vol. 332, pp. 6216‒6233, 2013.
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  18.  J. Tessarzik, Flexible rotor balancing by the influence coefficient method. Part 1: Evaluation of the exact point speed and least squares procedure. Latham: Mechanical Technology Incorporated, 1972.
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Authors and Affiliations

Georg Quinz
1
Marcel S. Prem
1
Michael Klanner
1
ORCID: ORCID
Katrin Ellermann
1

  1. Graz University of Technology, Institute of Mechanics, Kopernikusgasse 24/IV, 8010 Graz, Austria
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Abstract

Vibration in rotating machinery leads to a series of undesired effects, e.g. noise, reduced service life or even machine failure. Even though there are many sources of vibrations in a rotating machine, the most common one is mass unbalance. Therefore, a detailed knowledge of the system behavior due to mass unbalance is crucial in the design phase of a rotor-bearing system. The modelling of the rotor and mass unbalance as a lumped system is a widely used approach to calculate the whirling motion of a rotor-bearing system. A more accurate representation of the real system can be found by a continuous model, especially if the mass unbalance is not constant and arbitrarily oriented in space. Therefore, a quasi-analytical method called Numerical Assembly Technique is extended in this paper, which allows for an efficient and accurate simulation of the unbalance response of a rotor-bearing system. The rotor shaft is modelled by the Rayleigh beam theory including rotatory inertia and gyroscopic effects. Rigid discs can be mounted onto the rotor and the bearings are modeled by linear translational/rotational springs/dampers, including cross-coupling effects. The effect of a constant axial force or torque on the system response is also examined in the simulation.
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Bibliography

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  4.  A.-C. Lee and Y.-P. Shih, “The Analysis of Linear Rotor-Bearing Systems: A General Transfer Matrix Method,” J. Vib. Acoust., vol. 115, no. 4, pp. 490–497, 1993.
  5.  T. Yang and C. Lin, “Estimation of Distributed Unbalance of Rotors,” J. Eng. Gas Turbines Power, vol. 124, no. 4, pp. 976‒983, 2002.
  6.  J.-S. Wu and H.-M. Chou, “A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses,” J. Sound Vib., vol. 81, no. 3, pp.  1–10, 1999.
  7.  J.-S. Wu, F.-T. Lin, and H.-J. Shaw, “Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks,” J. Appl. Mech., vol. 220, no.  3, pp. 451–468, 2014.
  8.  M. Klanner and K. Ellermann, “Steady-state linear harmonic vibrations of multiple-stepped Euler-Bernoulli beams under arbitrarily distributed loads carrying any number of concentrated elements,” Appl. Comput. Mech., vol. 14, no. 1, pp. 31–50, 2020.
  9.  M. Klanner, M.S. Prem, and K. Ellermann, “Steady-state harmonic vibrations of a linear rotor-bearing system with a discontinuous shaft and arbitrary distributed mass unbalance,” in Proceedings of ISMA2020 International Conference on Noise and Vibration Engineering and USD2020 International Conference on Uncertainty in Structural Dynamics, Leuven, Belgium, Sep. 2020, pp. 1257–1272.
  10.  H. Ziegler, “Knickung gerader Stäbe unter Torsion,” J. Appl. Math. Phys. (ZAMP), vol. 3, pp. 96–119, 1952.
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  18.  A.-C. Lee, Y. Kang, and L. Shin-Li, “A Modified Transfer Matrix Method for Linear Rotor-Bearing Systems,” J. Appl. Mech., vol. 58, no. 3, pp. 776–783, 1991.
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  21.  R.L. Eshleman and R.A. Eubanks, “On the Critical Speeds of a Continuous Rotor,” J. Manuf. Sci. Eng., vol. 91, no. 4, pp. 1180‒1188, 1969.
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Authors and Affiliations

Michael Klanner
1
ORCID: ORCID
Marcel S. Prem
1
Katrin Ellermann
1

  1. Graz University of Technology, Institute of Mechanics, Kopernikusgasse 24/IV, 8010 Graz, Austria

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