Dynamics of structural systems with various frequency-dependent damping models

Abstract

The aim of this paper is to present the dynamic analyses of the system involving various damping models. The assumed frequency-dependent damping forces depend on the past history of motion via convolution integrals over some damping kernel functions. By choosing suitable damping kernel functions of frequency-dependent damping model, it may be derived from the familiar viscoelastic materials. A brief review of literature on the choice of available damping models is presented. Both the mode superposition method and Fourier transform method are developed for calculating the dynamic response of the structural systems with various damping models. It is shown that in the case of non-deficient systems with various damping models, the modal analysis with repeated eigenvalues are very similar to the traditional modal analysis used in undamped or viscously damped systems. Also, based on the pseudo-force approach, we transform the original equations of motion with nonzero initial conditions into an equivalent one with zero initial conditions and therefore present a Fourier transform method for the dynamics of structural systems with various damping models. Finally, some case studies are used to show the application and effectiveness of the derived formulas.

Keywords

damping / viscoelasticity / dynamic analysis / mode superposition method / Fourier transform method

References

[1] Rayleigh L. The Theory of Sound. New York: Dover Publications, 1877

[2] Caughey T K, O’Kelly M E J. Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics, 1965, 32(3): 583–588

[3] Adhikari S. Structural Dynamic Analysis with Generalized Damping Models: Analysis. Hoboken: John Wiley & Sons, 2013

[4] Liu C, Jing X, Daley S, Recent advances in micro-vibration isolation. Mechanical Systems and Signal Processing, 2015, 56–57(May): 55–80

[5] Li L, Hu Y. Generalized mode acceleration and modal truncation augmentation methods for the harmonic response analysis of nonviscously damped systems. Mechanical Systems and Signal Processing, 2015, 52–53(February): 46–59

[6] Bagley R L, Torvik P J. Fractional calculus— A different approach to the analysis of viscoelastically damped structures. AIAA Journal, 1983, 21(5): 741–748

[7] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Singapore: World Scientific, 2010

[8] Lewandowski R, Chorążyczewski B. Identification of the parameters of the Kelvin–Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers. Computers & Structures, 2010, 88(1–2): 1–17

[9] Di Paola M, Pinnola F P, Zingales M. Fractional differential equations and related exact mechanical models. Computers & Mathematics with Applications (Oxford, England), 2013, 66(5): 608–620

[10] Enelund M, Lesieutre G A. Time domain modeling of damping using anelastic displacement fields and fractional calculus. International Journal of Solids and Structures, 1999, 36(29): 4447– 4472

[11] Di Paola M, Pirrotta A, Valenza A. Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results. Mechanics of Materials, 2011, 43(12): 799–806

[12] Biot M A. Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. Journal of Applied Physics, 1954, 25(11): 1385–1391

[13] Biot M A. Variational principles in irreversible thermodynamics with application to viscoelasticity. Physical Review, 1955, 97(6): 1463–1469

[14] Adhikari S. Structural Dynamic Analysis with Generalized Damping Models: Identification. Hoboken: John Wiley & Sons, 2013

[15] Li L, Hu Y, Wang X. Design sensitivity analysis of dynamic response of nonviscously damped systems. Mechanical Systems and Signal Processing, 2013, 41(1–2): 613–638

[16] Li L, Hu Y, Wang X. Improved approximate methods for calculating frequency response function matrix and response of MDOF systems with viscoelastic hereditary terms. Journal of Sound and Vibration, 2013, 332(15): 3945–3956

[17] Zopf C, Hoque S, Kaliske M. Comparison of approaches to model viscoelasticity based on fractional time derivatives. Computational Materials Science, 2015, 98(February): 287–296

[18] Woodhouse J. Linear damping models for structural vibration. Journal of Sound and Vibration, 1998, 215(3): 547–569

[19] Golla D F, Hughes P C. Dynamics of viscoelastic structures-A time domain finite element formulation. Journal of Applied Mechanics, 1985, 52(4): 897–906

[20] McTavish D J, Hughes P C. Modeling of linear viscoelastic space structures. Journal of Vibration and Acoustics, 1993, 115(1): 103–110

[21] Lesieutre G A. Finite element modeling of frequency-dependent material damping using augmented thermodynamic fields. Journal of Guidance, Control, and Dynamics, 1990, 13(6): 1040–1050

[22] Lesieutre G A. Finite elements for dynamic modeling of uniaxial rods with frequency-dependent material properties. International Journal of Solids and Structures, 1992, 29(12): 1567–1579

[23] Renaud F, Dion J L, Chevallier G, A new identification method of viscoelastic behavior: Application to the generalized Maxwell model. Mechanical Systems and Signal Processing, 2011, 25(3): 991–1010

[24] Koeller R. Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 1984, 51(2): 299–307

[25] Adhikari S, Woodhouse J. Identification of damping: Part 1, viscous damping. Journal of Sound and Vibration, 2001, 243(1): 43–61

[26] García-Barruetabeía J, Cortés F, Manuel Abete J. Influence of nonviscous modes on transient response of lumped parameter systems with exponential damping. Journal of Vibration and Acoustics, 2011, 133(6): 064502

[27] Li L, Hu Y, Wang X, Eigensensitivity analysis for asymmetric nonviscous systems. AIAA Journal, 2013, 51(3): 738–741

[28] Adhikari S. Damping models for structural vibration. Dissertation for the Doctoral Degree. Cambridge: University of Cambridge, 2000

[29] Gonzalez-Lopez S, Fernandez-Saez J. Vibrations in Euler-Bernoulli beams treated with non-local damping patches. Computers & Structures, 2012, 108–109: 125–134

[30] Friswell M I, Adhikari S, Lei Y. Non-local finite element analysis of damped beams. International Journal of Solids and Structures, 2007, 44(22–23): 7564–7576

[31] Pan Y, Wang Y. Frequency-domain analysis of exponentially damped linear systems. Journal of Sound and Vibration, 2013, 332(7): 1754–1765

[32] Adhikari S, Friswell M I, Lei Y. Modal analysis of nonviscously damped beams. Journal of Applied Mechanics, 2007, 74(5): 1026–1030

[33] Palmeri A, Ricciardelli F, De Luca A, State space formulation for linear viscoelastic dynamic systems with memory. Journal of Engineering Mechanics, 2003, 129(7): 715–724

[34] Wagner N, Adhikari S. Symmetric state-space formulation for a class of non-viscously damped systems. AIAA Journal, 2003, 41(5): 951–956

[35] Palmeri A, Adhikari S. A Galerkin-type state-space approach for transverse vibrations of slender double-beam systems with visco-elastic inner layer. Journal of Sound and Vibration, 2011, 330(26): 6372–6386

[36] Tran Q H, Ouisse M, Bouhaddi N. A robust component mode synthesis method for stochastic damped vibroacoustics. Mechanical Systems and Signal Processing, 2010, 24(1): 164–181

[37] Di Paola M, Pinnola F P, Spanos P D. Analysis of multi-degree-of-freedom systems with fractional derivative elements of rational order. In: Proceedings of 2014 International Conference on Fractional Differentiation and Its Applications (ICFDA). IEEE, 2014, 1–6

[38] Adhikari S, Pascual B. Iterative methods for eigenvalues of viscoelastic systems. Journal of Vibration and Acoustics, 2011, 133(2): 021002

[39] Adhikari S, Pascual B. Eigenvalues of linear viscoelastic systems. Journal of Sound and Vibration, 2009, 325(4–5): 1000–1011

[40] Cortés F, Elejabarrieta M J. Computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments. Computer Methods in Applied Mechanics and Engineering, 2006, 195(44–47): 6448–6462

[41] Bilasse M, Charpentier I, Daya E M, A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method. Computer Methods in Applied Mechanics and Engineering, 2009, 198(49–52): 3999–4004

[42] Daya E, Potier-Ferry M. A numerical method for nonlinear eigenvalue problems application to vibrations of viscoelastic structures. Computers & Structures, 2001, 79(5): 533–541

[43] Lázaro M, Pérez-Aparicio J L, Epstein M. A viscous approach based on oscillatory eigensolutions for viscoelastically damped vibrating systems. Mechanical Systems and Signal Processing, 2013, 40(2): 767–782

[44] LáZaro M, Pérez–Aparicio J L. Multiparametric computation of eigenvalues for linear viscoelastic structures. Computers & Structures, 2013, 117(February): 67–81

[45] Lázaro M, Pérez-Aparicio J L. Dynamic analysis of frame structures with free viscoelastic layers: New closed-form solutions of eigenvalues and a viscous approach. Engineering Structures, 2013, 54(September): 69–81

[46] Pawlak Z, Lewandowski R. The continuation method for the eigenvalue problem of structures with viscoelastic dampers. Computers & Structures, 2013, 125(September): 53–61

[47] Van Beeumen R, Meerbergen K, Michiels W. A rational Krylov method based on Hermite interpolation for nonlinear eigenvalue problems. SIAM Journal on Scientific Computing, 2013, 35(1): A327–A350

[48] Güttel S, van Beeumen R, Meerbergen K, NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems. SIAM Journal on Scientific Computing, 2014, 36(6): A2842– A2864

[49] MSC. Software Corporation. MD/MSC Nastran 2010 Dynamic Analysis User’s Guide, 2010

[50] Li L, Hu Y, Wang X. Design sensitivity and Hessian matrix of generalized eigenproblems. Mechanical Systems and Signal Processing, 2014, 43(1–2): 272–294

[51] Murthy D V, Haftka R T. Derivatives of eigenvalues and eigenvectors of a general complex matrix. International Journal for Numerical Methods in Engineering, 1988, 26(2): 293–311

[52] Andrew A L. Convergence of an iterative method for derivatives of eigensystems. Journal of Computational Physics, 1978, 26(1): 107–112

[53] Diekmann O, van Gils S A, Lunel S V, Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Berlin: Springer-Verlag, 1995

[54] Li L, Hu Y, Wang X. Eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems. Journal of Sound and Vibration, 2014, 333(4): 1182–1192

[55] Li L, Hu Y, Wang X, A hybrid expansion method for frequency response functions of non-proportionally damped systems. Mechanical Systems and Signal Processing, 2014, 42(1–2): 31–41

[56] Adhiakri S. Classical normal modes in nonviscously damped linear systems. AIAA Journal, 2001, 39(5): 978–980

[57] Veletsos A S, Ventura C E. Dynamic analysis of structures by the DFT method. Journal of Structural Engineering, 1985, 111(12): 2625–2642

[58] Lee U, Kim S, Cho J. Dynamic analysis of the linear discrete dynamic systems subjected to the initial conditions by using an FFT-based spectral analysis method. Journal of Sound and Vibration, 2005, 288(1–2): 293–306

[59] Mansur W, Soares D Jr, Ferro M. Initial conditions in frequency-domain analysis: The FEM applied to the scalar wave equation. Journal of Sound and Vibration, 2004, 270(4–5): 767–780

[60] Barkanov E, Hufenbach W, Kroll L. Transient response analysis of systems with different damping models. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 33–46

[61] Brigham E O. The Fast Fourier Transform and Its Applications. Englewood Cliffs: Prentice-Hall, 1988

[62] Duhamel P, Vetterli M. Fast Fourier transforms: A tutorial review and a state of the art. Signal Processing, 1990, 19(4): 259–299

[64] Zghal S, Bouazizi M L, Bouhaddi N. Reduced-order model for non-linear dynamic analysis of viscoelastic sandwich structures in time domain. In: Proceedings of International Conference on Structure Nonlinear Dynamics and Diagnosis. EDP Sciences, 2014, 08003

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant No. 51375184).

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg