Skip to main content

Simplification of the forced response of mistuned bladed disks using multiple scales techniques

Abstract

Mistuning can produce a considerable increase of the vibratory forced response of the blades compared with that of the tuned bladed disk. This situation can lead to high cycle fatigue failure, and, therefore, it is of crucial importance for the prediction of the safe operation limit of the bladed disk. Nonlinear friction forces determine the final vibration amplitude of the blades, and, because of mistuning, the complete bladed disk has to be considered, increasing considerably the difficulty of the problem. For this reason, we consider the application of asymptotic multiple scale techniques to derive simplified models that give the final vibration states at a very low CPU cost. This asymptotic method is based on the idea that all significant effects (forcing, nonlinear friction damping, and mistuning) produce, in most practical situations, just small corrections of the tuned blade elastic oscillation. These small effects develop in a much longer time scale than that associated with the natural elastic frequency of the tuned system. The reduced models produced by this asymptotic methodology retain only the slow time dynamics, filtering out the fast scale oscillation. In this paper, the bladed disk is described using a mass-spring system with nonlinear friction and with an external forcing acting on a blade-dominated modal family where all modes have very similar vibration frequencies. The derivation of the asymptotic model from the mass-spring system is explained in detail, and validated against the results from the original mass-spring model.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

References

  1. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers: I: Asymptotic Methods and Perturbation Theory. Springer, New York (1999). https://doi.org/10.1007/978-1-4757-3069-2

  2. Corral, R., Gallardo, J.M.: A methodology for the vibration amplitude prediction of self-excited rotors based on dimensional analysis. In: Proceedings of ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition, ASME, GT2006-90668, pp 1101–1113 (2016) https://doi.org/10.1115/GT2006-90668

  3. Corral, R., Beloki, J., Calza, P., Elliott, R.: Flutter generation and control using mistuning in a turbine rotating rig. AIAA J. 57(2), 782–795 (2019). https://doi.org/10.2514/1.J056943

    Article  Google Scholar 

  4. Kevorkian, J., Cole, J.: Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences, vol 114. Springer, New York (1996). https://doi.org/10.1007/978-1-4612-3968-0

  5. Krack, M., Salles, L., Thouverez, F.: Vibration prediction of bladed disks coupled by friction joints. Arch. Comput. Methods Eng. 24, 589–636 (2017). https://doi.org/10.1007/s11831-016-9183-2

    Article  MATH  Google Scholar 

  6. Martel, C., Corral, R.: Asymptotic description of maximum mistuning amplification of bladed disk forced response. J. Eng. Gas Turbin. Power 131(2), 022506 (2009). https://doi.org/10.1115/1.2968868

    Article  Google Scholar 

  7. Martel, C., Martin, J.A.: Asymptotic description of forced response vibration saturation by friction forces. J. Eng. Gas Turbines Power 142(2), 011022:1–8 (2020). https://doi.org/10.1115/1.4044934

    Article  Google Scholar 

  8. Martel, C., Corral, R., Ivaturi, R.: Flutter amplitude saturation by nonlinear friction forces: Reduced model verification. J. Turbomach. 137(4), 041004:1–8 (2014). https://doi.org/10.1115/1.4028443

    Article  Google Scholar 

  9. Olofsson, U.: Cyclic microslip under unlubricated conditions. Tribol. Int. 28(4), 207–217 (1995). https://doi.org/10.1016/0301-679X28942900001-7

    Article  Google Scholar 

  10. Petrov, E.P., Ewins, D.J.: Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks. J. Turbomach. 125(2), 364–371 (2003). https://doi.org/10.1115/1.1539868

    Article  Google Scholar 

  11. Petrov, E.P., Ewins, D.J.: State-of-the-art dynamic analysis for non-linear gas turbine structures. Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 218(3), 199–211 (2004). https://doi.org/10.1243/0954410041872906

    Article  Google Scholar 

  12. Yang, B., Menq, C.: Characterization of 3d contact kinematics and prediction of resonant response of structures having a 3d frictional constraint. J. Sound Vib. 217(5), 909–925 (1998). https://doi.org/10.1006/jsvi.1998.1802

    Article  Google Scholar 

  13. Yang, B., Chu, M., Menq, C.: Stick-slip-separation analysis and non-linear stiffnes and damping characterization of friction contacts having variable normal load. J. Sound Vib. 210(4), 461–481 (1998). https://doi.org/10.1006/jsvi.1997.1305

    Article  Google Scholar 

Download references

Acknowledgements

This work has been supported by the Spanish Ministerio de Ciencia, Innovación y Universidades under Grant DPI2017-84700-R.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Juan A. Martín.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Martín, J.A., Martel, C. Simplification of the forced response of mistuned bladed disks using multiple scales techniques. Nonlinear Dyn 104, 2037–2049 (2021). https://doi.org/10.1007/s11071-021-06397-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06397-0

Keywords

  • Asymptotic method
  • Blade vibration
  • Nonlinear friction
  • Turbomachinery