Analysis of dynamic behaviour of viscoelastic rods whose rheological models contain fractional derivatives of two different orders

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 81, Issue 6, 2001, Pages 363-376

  Rossikhin, Yu A ^a   Shitikova, M V ^a  

^a VORONEZH STATE TECHNICAL UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0347142866     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/1521-4001(200106)81:6<363::AID-ZAMM363>3.0.CO;2-9     Document Type: Article

References (24)

1. BAGLEY, R. L.: Applications of generalized derivatives to viscoelasticity. Ph.D. Dissertation. Air Force Institute of Technology, 1979.
2. BAGLEY, R. L.; TORVIK, P. J.: Fractional calculus - a different approach to the analysis of viscoelastically damped structures. AIAA J. 21 (1983) 5, 741-748.
3. FRIEDRICH, CH.: Relaxation functions of rheological constitutive equations with fractional derivatives: thermodynamic constraints. In: CASAS-VAZQUES, J. (ed.): Rheological modelling: Thermodynamical and statistical approaches. Lecture Notes Phys. 381. Springer Verlag, Berlin, Heidelberg 1991, pp. 321-330.
4. FRIEDRICH, CH.: Mechanical stress relaxation in polymers: fractional integral model versus fractional differential model. J. Non-Newtonian Fluid Mech. 46 (1993), 307-314.
5. KEMPFLE, S.: Causality criteria for solutions of linear fractional differential equations. Fractional Calculus Appl. Analysis 1 (1998) 4, 351-364.
6. HAVRILIAK, S.; NEGAMI, S.: Analysis of α-dispersion in some polymer systems by the complex variables method. J. Polymer Science (C) 14 (1966), 99-117.
7. HAVRILIAK, S.; NEGAMI, S.: On the equivalence of dielectric and mechanical dispersions in some polymers; e.g. poly(n-octyl methacrylate). Polymer 10 (1966) 10, 859-872.
8. MORGENTHALER, D. R.: Practical design and analysis of systems with fractional derivative materials and active control. Proc. Damping'91, San Diego, CA, 1 (1991), BCA1-BCA28.
9. HARTMANN, B.; LEE, G. F.; LEE, J. D.: Loss factor height and width limits for polymer relaxations. J. Acoust. Soc. Amer. 95 (1994) 1, 226-233.
10. HORR, A. M.; SCHMIDT, L. C.: Dynamic response of a damped large space structure: a new fractional-spectral approach. Int. J. Space Struct. 2 (1995) 10, 113-120.
11. HORR, A. M.; SCHMIDT, L. C.: Frequency domain dynamic analysis of large space structures with added elastomeric dampers. Int. J. Space Struct. 11 (1996) 3, 279-289.
12. HORR, A. M.; SCHMIDT, L. C.: Fractional-spectral method for vibration of damped space structures. Eng. Struct. 18 (1996) 12, 947-956.
13. ROSSIKHIN, YU. A.; SHITIKOVA, M. V.: Application of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Reviews 50 (1997) 1, 15-67.
14. ROSSIKHIN, YU. A.; SHITIKOVA, M. V.: Vibrations of a hereditarily elastic mechanical system whose hereditary properties are described by fractional derivatives. Appl. Mech. Amer. 8 (1999) (Proc. 6th Pan American Congr. Appl. Mech., Rio de Janeiro, Jan. 4-8, 1999), pp. 1409-1412.
15. ROSSIKHIN, YU. A.; SHITIKOVA, M. V.: A new method for solving dynamic problems of fractional derivative viscoelasticity. Int. J. Engin. Sci. (to appear).
16. BLAND, D. R.: Theory of linear viscoelasticity. Pergamon, New York 1960.
17. GEL'FAND, I. M.; SHILOV, G. E.: Generalized functions. Vol. 1: Properties and operations. Academic Press, New York 1964.
18. CHRISTENSEN, R. M.: Theory of viscoelasticity: An introduction. Academic Press, New York 1971.
19. JEFFREYS, H.; SWIRLES, B.: Methods of mathematical physics. 3d ed. Cambridge University Press, Cambridge 1966.
20. ROSSIKHIN, YU. A.; SHITIKOVA, M. V.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single-mass systems. Acta Mech. 120 (1997) 1-4, 109-125.
21. GAUL, L.; SCHANZ, M.: Dynamics of viscoelastic solids treated by boundary element approaches in time domain. European J. Mech. A/Solids 13 (1994), 43-59.
22. GAUL, L.: The influence of damping on waves and vibrations. Mech. Systems Signal Processing 13 (1999), 1-30.
23. LAVRENTJEV, M. A.; SHABAT, B. V.: Methods of the theory of functions of complex variable. Nauka, Moscow 1965 (in Russian).
24. BATEMAN, H.; ERDELYI, A.: Tables of integral transforms 1. McGraw-Hill, New York 1954.