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A modal strategy devoted to the hidden state variables method with large interfaces

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Abstract

In many mechanical engineering applications, the interactions of a structure through its boundary is modelled by a dynamic boundary stiffness matrix. Nevertheless, it is well known that the solution of such computational model is very sensitive to the modelling uncertainties on the dynamic boundary stiffness matrix. In a recent work, the “hidden state variables method” is used to identify mass, stiffness and damping matrices associated with a given deterministic dynamic boundary stiffness matrix which can be constructed by using experimental measurements. Such an identification allows the construction of the probabilistic model of a random boundary stiffness matrix by substituting those identified mass, stiffness and damping matrices by random matrices. Nevertheless, the numerical cost of the “hidden state variables method” increases drastically with the dimension (number of degrees of freedom) of the interface. We then propose an enhanced approach which consists in a truncated spectral representation of the displacements on the boundary and with a partition of the frequency band of analysis. A collection of mass, stiffness and damping matrices is then identified for each sub-frequency band of analysis. A probabilistic model is constructed in substituting each of those matrices by random matrices. A numerical application is proposed.

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References

  1. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method, vol 1, 6th edn. Butterworth-Heinemann, Oxford

    Google Scholar 

  2. Givoli D (1992) Numerical methods for problems in infinite domains. Studies in applied mechanics. Elsevier, Amsterdam

    Google Scholar 

  3. Wolf JP, Song C (1996) Finite-element modelling of unbounded media. Wiley, Chichester

    Google Scholar 

  4. Turkel E (1998) Special issue on absorbing boundary conditions. Appl Numer Math 27(4):327–560

    Article  MathSciNet  Google Scholar 

  5. Magoulès F, Harari I (2006) Absorbing boundary conditions. Comput Methods Appl Mech Eng 195(29–32):3551–3902

    Article  Google Scholar 

  6. Tsynkov SV (1998) Numerical solution of problems on unbounded domains. a review. Appl Numer Math 27:465–532

    Article  MathSciNet  Google Scholar 

  7. Bonnet M (1999) Boundary integral equation methods for solids and fluids. Wiley-Blackwell, New York

    Google Scholar 

  8. Wolf JP, Song C (2001) The scaled boundary finite-element method—a fundamental solution-less boundary-element method. Comput Methods Appl Mech Eng 190(42):5551–5568

    Article  Google Scholar 

  9. Mondot J, Petersson B (1987) Characterization of struture-borne sound sources: The source descriptor and the coupling function. J Sound Vib 114:507–518

    Article  Google Scholar 

  10. Koh Y, White R (1996) Analysis and control of vibration power transmission to machinery supporting structures subjected to a multi-excitation system, Part I: Driving point mobility matrix of beams and rectangular plates. J Sound Vib 196:469–493

    Article  Google Scholar 

  11. Villot M, Ropars P, Bongini E et al (2011) Modelling the influence of structural modifications on the response of a building to railway vibration. Noise Control Eng J 59:641–651

    Article  Google Scholar 

  12. Cherukuri A, Barbone PE (1998) High modal density approximation for equipment in the time domain. J Acoust Soc Am 104(2):2048–2053

    Article  Google Scholar 

  13. Soize C (2000) A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab Eng Mech 15:277–294

    Article  Google Scholar 

  14. Soize C (2005) Random matrix theory for modeling uncertainties in computational mechanics. Comput Methods Appl Mech Eng 194:1333–1366

    Article  MathSciNet  Google Scholar 

  15. Cottereau R, Clouteau D, Soize C (2007) Construction of a probabilistic model for impedance matrices. Comput Methods Appl Mech Engrg 196:2252–2268

    Article  MathSciNet  Google Scholar 

  16. Cottereau R, Clouteau D, Soize C (2006) Probabilistic nonparametric model of impedance matrices: application to the seismic design of a structure. Eur J Comput Mech 15:131–142

    Google Scholar 

  17. Ropars P, Desceliers C, Jean P (2014) Quantification of uncertainties in a computational model of interaction soil-structure with a railway excitation. submit on October 2014 in J Vib Acoust

  18. Chabas F, Soize C (1987) Modeling mechanical subsystems by boundary impedance in the finite element method. La Rech Aérospa (english edition) 5:59–75

    MathSciNet  Google Scholar 

  19. Ropars P, Bonnet G, Jean P (2014) A stabilization process applied to a hidden variables method for evaluating the uncertainties on foundation impedances and their effect on vibrations induced by railways in a building. J Sound Vib 333(1):1953–1971

    Article  Google Scholar 

  20. Rayleigh JWS (1945) The theory of sound, vol 1. Diver Publications, New York

    Google Scholar 

  21. Dienstfrey A, Greengard L (2001) Analytic continuation, singular-value expansions, and Kramers–Kronig analysis. Inverse Probl 17:1307–1320

    Article  MathSciNet  Google Scholar 

  22. Cottereau R, Clouteau D, Soize C (2007) Modèle dynamique équivalent de matrices d’impédance de fondation. \(7^{\grave{{\rm e}}{\rm me}}\) Colloque National, AFPS 2007 - École Centrale Paris

  23. Friswell MI (1990) Candidate reduced order models for structural parameter estimation. J Vib Acoust 112:93–97

    Article  Google Scholar 

  24. Craig RR (1995) Substructure methods in vibration. J Vib Acoust 117(3):207–213

    Article  Google Scholar 

  25. Chaillat S, Bonnet M (2014) A new fast multipole formulation for the elastodynamic half-space green’s tensor. J Comput Phys 258:787–808

    Article  MathSciNet  Google Scholar 

  26. Bazyar MH, Song C (2008) A continued-fraction-based high-order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry. Int J Numer Methods Eng 74:209–237

    Article  Google Scholar 

  27. Tang S, Hou TY, Liu WK (2006) A mathematical framework of the bridging scale method. Int J Numer Methods Eng 65:1688–1713

    Article  MathSciNet  Google Scholar 

  28. Barbone PE, Cherukuri A, Goldman D (2000) Canonical representation of complex vibratory subsystems: time domain Dirichlet to Neumann maps. Int J Solids Struct 37:2825–2857

    Article  MathSciNet  Google Scholar 

  29. Barbone PE, Givoli D, Patlashenko I (2003) Optimal modal reduction of vibrating substructures. Int J Numer Methods Eng 57:341–369

    Article  MathSciNet  Google Scholar 

  30. Givoli D, Barbone PE, Patlashenko I (2004) Which are the important modes of a subsystem? Int J Numer Methods Eng 59:1657–1678

    Article  MathSciNet  Google Scholar 

  31. Sanathanan C, Koerner J (1963) Transfer function synthesis as a ratio of two complex polynomials. IEEE Trans Autom Control 8:56–58

    Article  Google Scholar 

  32. Du X, Zhao M (2009) Stability and identification for rational approximation of frequency response function of unbounded soil. Earthq Eng Struct Dyn 39:165–186

    Google Scholar 

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Authors and Affiliations

  1. MSME Université Paris-Est Marne-la-Vallée, 5 Bd Descartes, 77454, Marne-la-Vallée cedex 2, France

    Pierre Ropars & Christophe Desceliers

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  1. Pierre Ropars
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  2. Christophe Desceliers
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Correspondence to Pierre Ropars.

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Ropars, P., Desceliers, C. A modal strategy devoted to the hidden state variables method with large interfaces. Comput Mech 55, 805–818 (2015). https://doi.org/10.1007/s00466-015-1148-z

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  • DOI: https://doi.org/10.1007/s00466-015-1148-z

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