Nonlinear vibrations of viscoelastic plates of fractional derivative type: An AEM solution

Open Mechanics Journal

Volumn 4, Issue 1, 2010, Pages 8-20

  Babouskos, Nick G ^a   Katsikadelis, John T ^a  

^a NATIONAL TECHNICAL UNIVERSITY OF ATHENS   (Greece)

Author keywords

Analog equation method; Boundary elements; Fractional partial differential equations; Fractional viscoelastic models; Nonlinear resonance; Nonlinear vibrations; Numerical solution; Plates; Viscoelasticity

Indexed keywords

EID: 77956190856     PISSN: None     EISSN: 18741584     Source Type: Journal    
DOI: 10.2174/1874158401004010008     Document Type: Article

References (31)

1. Stiassnie M. On the application of fractional calculus for the formulation of viscoelastic models. Appl Math Model 1979; 3: 300-02.
2. Haneczok G, Weller M. A fractional model of viscoelastic relaxation. Mater Sci Eng A 2004; 370: 209-12.
3. Ramirez LES, Coimbra CFM. A variable order constitutive relation for viscoelasticity. Ann Phys (Leipzig) 2007; 16: 543-52.
4. Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley 1979.
5. Fang T, Dowell EH. Numerical simulation of jump phenomena in stable Duffing systems. Int J Non-Linear Mech 1987; 22: 267-74.
6. Chu H-N, Herrmann G. Influence of large amplitude on free flexural vibrations of rectangular elastic plates. J Appl Mech 1956; 23: 532-40.
7. Sridhar S, Mook DT, Nayfeh AH. Non-Linear resonances in the forced responses of plates, Part I: symmetric responses of circular plates. Sound Vib 1975; 41: 359-73.
8. Amabili M. Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Comp Struct 2004; 82: 2587-650.
9. Parzygnat WJ, Pao Y-H. Resonance phenomena in the nonlinear vibration of plates governed by Duffing's equation. Int J Eng Sci 1978; 16: 999-1017.
10. Haterbouch M, Benamar R. Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates. J Sound Vib 2005; 280: 903-24.
11. Leung AYT, Mao SG. A symplectic Galerkin method for nonlinear vibration of beams and plates. J Sound Vib 1995; 183: 475-91.
12. Lee Y-Y, Ng CF. Nonlinear response of plates using finite element modal reduction method. Eng Struct 2001; 23: 1104-14.
13. Ribeiro P, Petyt M. Geometrical non-linear, steady state, forced, periodic vibration of plates, part I: model and convergence studies. J Sound Vib 1999; 226: 955-83.
14. Ribeiro P, Petyt M. Nonlinear vibration of plates by the hierarchical finite element and continuation methods. Int J Mech Sci 1999; 41: 437-59.
15. Wang X-X, Qian J, Huang M-K. A boundary integral equation formulation for large amplitude nonlinear vibration of thin elastic plates. Comput Methods Appl Mech Eng 1991; 86: 73-86.
16. Katsikadelis JT, Nerantzaki MS, Kandilas CB. A BEM approach to non-linear vibrations of plates. In: Moan T, Kounadis AN, Krätzig WB, Schuëller GI, Eds. Structural Dynamics-EURODYN '93. Proc. of the 2nd European Conference on Structural Dynamics: EURODYN 93; 1993 June 21-23. Trondheim, Norway, Balkema: Rotterdam 1992; pp. 659-71.
17. Nerantzaki MS, Katsikadelis JT. Nonlinear dynamic analysis of circular plates with varying thickness. Arch Appl Mech 2007; 77: 381-91.
18. Ioannides E, Grootenhuis P. A finite element analysis of the harmonic response of damped three-layer plates. J Sound Vib 1979; 67: 203-18.
19. Eshmatov BKh. Nonlinear vibrations and dynamic stability of viscoelastic orthotropic rectangular plates. J Sound Vib 2007; 300: 709-26.
20. Khodzhaevl DA, Eshmatov BKh. Nonlinear vibrations of a viscoelastic plate with concentrated masses. J Appl Mech Tech Phys 2007; 48: 905-14.
21. Rossikhin YuA, Shitikova MV. Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances. Int J Non-Linear Mech 2006; 41: 313-25.
22. Katsikadelis JT. The analog boundary integral equation method for nonlinear static and dynamic problems in continuum mechanics. J Theor Appl Mech 2002; 40: 961-84.
23. Katsikadelis JT. Numerical solution of multi-term fractional differential equations. ZAMM Zeitschrift für Angewandte Math Mech 2009; 89: 593-608.
24. Gaul L. The influence of damping on waves and vibrations. Mech Syst Signal Proc 1999; 13: 1-30.
25. Podlubny I. Fractional differential equations. New York: Academic Press 1999.
26. Malvern LE. Introduction to the Mechanics of a continuum medium. Engelwood Cliffs, New Jersey: Prentice-Hall 1969.
27. Katsikadelis JT, Babouskos NG. Nonlinear flutter instability of thin damped plates. A solution by the analog equation method. J Mech Mater Struct 2009; 4: 1395-414.
28. Katsikadelis JT. Special methods for plate analysis. In: Beskos DE, Ed. Boundary element analysis of plates and shells. New York: Springer-Verlag 1991; pp. 221-311.
29. Thompson JF, Soni BK, Weatherill NP. Handbook of grid generation. USA: CRC Press 1998.
30. Katsikadelis JT. Boundary elements: theory and applications. Amsterdam-London: Elsevier 2002.
31. Amabili M. Nonlinear vibrations and stability of shells and plates. USA: Cambridge University Press 2008.