Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. (English) Zbl 0901.73030

(Author’s summary.) We study damped vibrations of an oscillator, whose viscoelastic properties are described in terms of the fractional calculus applied to Kelvin-Voigt model, Maxwell model, and standard linear solid model. The problem is solved by the Laplace transform method. When passing from image to original, one has to find the roots of an algebraic equation with fractional exponents. A method for solving this equation is proposed which allows one to investigate the roots behaviour in a wide range of single-mass system parameters. A comparison between the results obtained on the basis of the three models has been carried out. It has been shown that for all models the characteristic equations do not possess real roots, but have one pair of complex conjugates, i.e., the test single-mass systems subjected to the impulse excitation do not pass into an aperiodic regime for any magnitudes of the relaxation and creep times. Main characteristics of vibratory motions of the single-mass system as functions of the relaxation time or creep time, which are equivalent to the temperature dependencies, are constructed and analyzed for all three models.


74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
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