### ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01

## 2aSA11. Applications of fractional calculus to the analysis of damped
vibrations of viscoelastic oscillators.

**Yuriy A. Rossikhin
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*Dept. of Theoretical Mechanics, Voronezh State Acad. of Construction and
Architecture, ul.Kirova 3-75, Voronezh 394018, Russia
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**Marina V. Shitikova
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*Voronezh State Acad. of Construction and Architecture, Voronezh 394018,
Russia
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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.

All posters will be on display from 9:00 a.m. to 12:00 noon. To allow
contributors an opportunity to see other posters, contributors of odd-numbered
papers will be at their posters from 9:00 a.m. to 10:30 a.m. and contributors of
even-numbered papers will be at their posters from 10:30 a.m. to 12:00 noon. To
allow for extended viewing time, posters will remain on display until 7:30 a.m.
Wednesday morning.