Nonlinear Dynamics

, Volume 88, Issue 1, pp 223–237

# An effective numerical method for calculating nonlinear dynamics of structures with dry friction: application to predict the vibration response of blades with underplatform dampers

• Dayi Zhang
• Jianwei Fu
• Qicheng Zhang
• Jie Hong
Original Paper

## Abstract

This paper describes an efficient method to predict the nonlinear steady-state response of a complex structure with multi-scattered friction contacts. The contact friction force is equivalent to additional stiffness and damping based on optimal approximation theory, and as a consequence, the computation is simplified greatly by the linearization for a nonlinear system. In order to obtain accurate pressure distribution on the contact interfaces, the dynamic contact normal pressure is obtained by the equivalent static analysis which is validated for most engineering cases. Considering the complex procedure to determine the transformation between two different contact states, the differential forms of friction force are given to solve the tangential force accurately under the complex movement of interfaces. The approaches developed in this paper are particularly suitable to solve the dynamic response of large-scale structures with local contact nonlinearities. The entire procedure to calculate the steady-state response of a finite element model with a large number of degrees of freedom is demonstrated taking the blades with underplatform dampers as an example. The method is proved to be accurate and efficient; in particular, it does not suffer convergence problem in the allowable range of precision error, which exhibits remarkable potential engineering application values.

## Keywords

Dry friction Equivalent stiffness and damping Underplatform damper Nonlinear dynamics

## Notes

### Acknowledgements

This work was financially supported by the National Natural Science Foundation of China, the Grant Nos. 51475021 and 51575022.

## References

1. 1.
Ma, H., Yin, F., Wu, Z., Tai, X., Wen, B.: Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing. Nonlinear Dyn. 84(3), 1225–1258 (2016)
2. 2.
Pennestrì, E., Rossi, V., Salvini, P., Valentini, P.P.: Review and comparison of dry friction force models. Nonlinear Dyn. 83(4), 1785–1801 (2016)
3. 3.
Pascal, M.: Sticking and nonsticking orbits for a two-degree-of-freedom oscillator excited by dry friction and harmonic loading. Nonlinear Dyn. 77(1), 267–276 (2014)
4. 4.
Jeffrey, M.R.: On the mathematical basis of solid friction. Nonlinear Dyn. 81(4), 1699–1716 (2015)
5. 5.
Ma, H., Wang, D., Tai, X., Wen, B.: Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition. J. Vib. Control (2015). doi:
6. 6.
Yang, B.D., Chu, M.L., Menq, C.H.: Stick-slip-separation analysis and non-linear stiffness and damping characterization of friction contacts having variable normal load. J. Sound Vib. 210(4), 461–481 (1998)
7. 7.
Avalos, J., Mignolet, M.P.: On damping entire bladed disks through dampers on only a few blades. J. Eng. Gas Turbines Power 132(9), 092503 (2010)
8. 8.
Laxalde, D., Thouverez, F., Lombard, J.P.: Forced response analysis of integrally bladed disks with friction ring dampers. J. Vib. Acoust. 132(1), 011013 (2010)
9. 9.
Hohl, A., Siewert, C., Panning, L., Kayser, A.: Nonlinear vibration analysis of gas turbine bladings with shroud coupling. In: ASME Turbo Expo, vol. 5, pp. 425–433 (2008)Google Scholar
10. 10.
Zucca, S., Firrone, C.M.: Nonlinear dynamics of mechanical systems with friction contacts: coupled static and dynamic multi-harmonic balance method and multiple solutions. J. Sound Vib. 333(3), 916–926 (2014)
11. 11.
Petrov, E.P.: Method for direct parametric analysis of nonlinear forced response of bladed discs with friction contact interfaces. In: ASME Turbo Expo, vol. 6, pp. 397–408 (2004)Google Scholar
12. 12.
Sanliturk, K.Y., Ewins, D.J.: Modelling two-dimensional friction contact and its application using harmonic balance method. J. Sound Vib. 193(2), 511–523 (1996)
13. 13.
Petrov, E.P., Ewins, D.J.: Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed discs. In: ASME Turbo Expo, vol. 4, pp. 499–908 (2002)Google Scholar
14. 14.
Menq, C.H., Yang, B.D.: Non-linear spring resistance and friction damping of frictional constraint having two-dimensional motion. J. Sound Vib. 217(1), 127–143 (1998)
15. 15.
Armstrong-Helouvry, B., Dupont, P., De Wit, C.C.: A survey of models, analysis tools, and compensation method for the control of machines with friction. Automatica 30(7), 1083–1138 (1994)
16. 16.
Dahl, P.R.: A Solid Friction Model. The Aerospace Corporation, EL Segundo, CA (1986)Google Scholar
17. 17.
Hornstein, A.: Dynamical modeling with application to friction phenomena. Ph.D. thesis, Göttingen-Deutschland (2005)Google Scholar
18. 18.
Sanliturk, K.Y., Imregun, M., Ewins, D.J.: Harmonic balance vibration analysis of turbine blades with friction dampers. J. Vib. Acoust. 119(1), 96–103 (1997)
19. 19.
Firrone, C.M., Zucca, S., Gola, M.: Effect of static/dynamic coupling on the forced response of turbine bladed disks with underplatform dampers. In: ASME Turbo Expo, vol. 6, pp. 429–440 (2009)Google Scholar
20. 20.
Chen, J.J., Yang, B.D., Menq, C.H.: Periodic forced response of structures having three-dimensional frictional constraints. J. Sound Vib. 229(4), 775–792 (2000)
21. 21.
Ciğeroğlu, E., Özgüven, H.N.: Nonlinear vibration analysis of bladed disks with dry friction dampers. J. Sound Vib. 295(3), 1028–1043 (2006)Google Scholar
22. 22.
Wang, A., Long, Q.: Forced response characteristics of bladed disks with mistuning non-linear friction. J. Cent. South Univ. 18, 679–684 (2011)
23. 23.
Schwingshackl, C.W., Petrov, E.P., Ewins, D.J.: Effects of contact interface parameters on vibration of turbine bladed disks with underplatform dampers. J. Eng. Gas Turbines Power 134(3), 032507 (2012)Google Scholar
24. 24.
Panning, L., Popp, K., Sextro, W., Götting, F., Kayser, A., Wolter, I.: Asymmetrical underplatform dampers in gas turbine bladings: theory and application. In: ASME Turbo Expo, vol. 6, pp. 269–280 (2004)Google Scholar
25. 25.
Zucca, S., Firrone, C.M., Gola, M.: Coupled static/dynamic modeling of wedge dampers for turbine blades. In: ASME Turbo Expo, vol. 6, pp. 1073–1086 (2010)Google Scholar
26. 26.
J Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)Google Scholar
27. 27.
Popov, V.L., Heß, M.: Method of Dimensionality Reduction in Contact Mechanics and Friction. Springer, Berlin (2015)
28. 28.
Dundurs, J.: Properties of elastic bodies in contact. In: de Pater, A.D., Kalker, J.J. (eds.) The Mechanics of the Contact Between Deformable Bodies, pp. 54–66. Delft University Press, Delft (1975)Google Scholar

## Authors and Affiliations

• Dayi Zhang
• 1
• Jianwei Fu
• 1
• Qicheng Zhang
• 1
• Jie Hong
• 1
• 2
1. 1.School of Energy and Power EngineeringBeihang UniversityBeijingPeople’s Republic of China
2. 2.Collaborative Innovation Center of Advanced Aero-EngineBeijingPeople’s Republic of China