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Nonlinear modal analysis of mechanical systems with frictionless contact interfaces

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Abstract

This paper explores mechanical systems undergoing unilateral frictionless contact conditions in the framework of nonlinear modal analysis. The nonlinear eigenproblem is formulated in the frequency domain through the minimization of a Rayleigh quotient subject to non-penetration inequality constraints. An additional equality constraint is introduced for normalization purposes. The resulting constrained minimization problem is then solved using an augmented Lagrangian strategy. Two applications are proposed: a thin longitudinal rod in unilateral contact with a rigid obstacle and a turbomachinery compressor blade in contact with a rigid casing. The first application illustrates the complexity of the nonlinear modal characterization of a system experiencing unilateral contact conditions while the second demonstrates the applicability of the proposed approach to large-scale mechanical systems involving non-smooth nonlinear terms.

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References

  1. Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng 92(3): 353–375. doi:10.1016/0045-7825(91)90022-X

    Article  MATH  MathSciNet  Google Scholar 

  2. Arora JS, Chahande AI, Paeng JK (1991) Multiplier methods for engineering optimization. Int J Numer Methods Eng 32(7): 1485–1525. doi:10.1002/nme.1620320706

    Article  MATH  MathSciNet  Google Scholar 

  3. Arquier R, Bellizzi S, Bouc R, Cochelin B (2006) Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes. Comput Struct 84(24–25): 1565–1576. doi:10.1016/j.compstruc.2006.01.011

    Article  MathSciNet  Google Scholar 

  4. Azeez MFA, Vakakis AF (1998) Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts. Int J Non-Linear Mech 34(3): 415–435. doi:10.1016/S0020-7462(98)00022-5

    Article  Google Scholar 

  5. Bellizzi S, Bouc R (2005) A new formulation for the existence and calculation of nonlinear normal modes. J Sound Vib 287(3): 545–569. doi:10.1016/j.jsv.2004.11.014

    MathSciNet  Google Scholar 

  6. Carpenter NJ, Taylor RL, Katona MG (1991) Lagrange constraints for transient finite element surface contact. Int J Numer Methods Eng 32(1): 103–128. doi:10.1002/nme.1620320107

    Article  MATH  Google Scholar 

  7. Jones E, Oliphant T, Peterson P et al (2001) Scipy: Open source scientific tools for python. URL http://www.scipy.org/

  8. Kerschen G, Peeters M, Golinval J, Vakakis A (2009) Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech Syst Signal Process 23(1): 170–194. doi:10.1016/j.ymssp.2008.04.002

    Article  Google Scholar 

  9. Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multibody, large deformation-frictional contact problems. Int J Numer Methods Eng 36(20): 3451–3485. doi:10.1002/nme.1620362005

    Article  MATH  MathSciNet  Google Scholar 

  10. Laxalde D, Thouverez F (2009) Complex non-linear modal analysis for mechanical systems: Application to turbomachinery bladings with friction interfaces. J Sound Vib 322(4–5): 1009–1025. doi:10.1016/j.jsv.2008.11.044

    Article  Google Scholar 

  11. Legrand M, Pierre C, Cartraud P, Lombard JP (2009) Two-dimensional modeling of an aircraft engine structural bladed disk-casing modal interaction. J Sound Vib 319(1–2): 366–391. doi:10.1016/j.jsv.2008.06.019

    Article  Google Scholar 

  12. Leib N, Nacivet S, Thouverez F, Jézéquel L (2007) Experimental and numerical analysis of two vibro-impacting beams. In: Proceedings of international modal analysis conference (IMAC) XXVI, Orlando, Florida

  13. Lesaffre N, Sinou JJ, Thouverez F (2007) Stability analysis of rotating beams rubbing on an elastic circular structure. J Sound Vib 299(4–5): 1005–1032. doi:10.1016/j.jsv.2006.08.027

    Article  Google Scholar 

  14. Mijar A, Arora JS (2000) Review of formulations for elastostatic frictional contact problems. Struct Multidiscip Optim 20(3): 167–189. doi:10.1007/s001580050147

    Article  Google Scholar 

  15. Peeters M, Viguié R, Sérandour G, Kerschen G, Golinval JC (2009) Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques. Mech Syst Signal Process 23(1): 195–216. doi:10.1016/j.ymssp.2008.04.003

    Article  Google Scholar 

  16. Pesheck E, Pierre C, Shaw SW (2002) A new galerkin-based approach for accurate non-linear normal modes through invariant manifolds. J Sound Vib 249(5): 971–993. doi:10.1006/jsvi.2001.3914

    Article  MathSciNet  Google Scholar 

  17. Pietrzak G, Curnier A (1999) Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Comput Methods Appl Mech Eng 177(3–4): 351–381. doi:10.1016/S0045-7825(98)00388-0

    Article  MATH  MathSciNet  Google Scholar 

  18. Powell MJD (1978) Algorithms for nonlinear constraints that use lagrangian functions. Math Program 14(1): 224–248. doi:10.1007/BF01588967

    Article  MATH  Google Scholar 

  19. Vakakis AF, Manevitch LI, Mikhlin YV, Pilipchuk VN, Zevin AA (1996) Normal Modes and Localization in Nonlinear Systems. Wiley Series in Nonlinear Science. doi:10.1002/9783527617869

  20. Woo KC, Rodger AA, Neilson RD, Wiercigroch M (2000) Application of the harmonic balance method to ground moling machines operating in periodic regimes. Chaos Solitons Fractals 11(15): 2515–2525. doi:10.1016/S0960-0779(00)00075-8

    Article  MATH  Google Scholar 

  21. Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, Heidelberg. doi:10.1007/978-3-540-32609-0

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Authors and Affiliations

  1. Structural Dynamics and Vibration Laboratory, Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Denis Laxalde & Mathias Legrand

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  1. Denis Laxalde
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  2. Mathias Legrand
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Correspondence to Denis Laxalde.

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Laxalde, D., Legrand, M. Nonlinear modal analysis of mechanical systems with frictionless contact interfaces. Comput Mech 47, 469–478 (2011). https://doi.org/10.1007/s00466-010-0556-3

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  • DOI: https://doi.org/10.1007/s00466-010-0556-3

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