Yuriy A. Rossikhin
Dept. of Theoretical Mechanics, Voronezh State Acad. of Construction and Architecture, ul.Kirova 3-75, Voronezh 394018, Russia
Marina V. Shitikova
Voronezh State Acad. of Construction and Architecture, Voronezh 394018, Russia
Free damped vibrations of an oscillator, whose viscoelastic properties are described in terms of the fractional calculus Kelvin--Voight, Maxwell, standard linear solid models, and the models with a fractional operator, are determined. The problem is solved by the Laplace transform method. When passing from image to preimage, one is led to find the roots of an algebraic equation with fractional exponents. The method for solving such equations is proposed, which allows one to investigate the roots' behavior in a wide range of single-mass system parameters. A comparison between the results obtained on the basis of the four 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 in any of the magnitudes of the relaxation and creep times. Once more, a model with some fractional operator is constructed, which allows the oscillator to be both in the vibrating motion and in the aperiodic regime depending on the intervals over which the relaxation times for the given model vary, as well as on the order of fractional power and the ratio of relaxed to nonrelaxed modulus.
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