Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems

TLDR: In this paper, the Laplace transform method is used to find the roots of algebraic equations with fractional exponents, which allows one to investigate the roots behavior in a wide range of single-mass system parameters.

Abstract: Free damped vibrations of an oscillator, whose viscoelastic properties are described in terms of the fractional calculus Kelvin-Voight model, Maxwell model, and standard linear solid model are determined. The problem is solved by the Laplace transform method. When passing from image to pre-image 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 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 in none of 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.