A new method for solving dynamic problems of fractional derivative viscoelasticity

• Published in: International journal of engineering science 2001, Vol.39 (2), p.149-176
• Main Authors: Rossikhin, Yu.A., Shitikova, M.V.
• Format: Article
• Language: English
• Subjects: Dynamic problems; Fractional calculus; Viscoelasticity
• ISSN: 0020-7225; 1879-2197
• DOI: 10.1016/S0020-7225(00)00025-2

Nonstationary vibrations of linear viscoelastic one-degree-of-freedom (1dof), two-degree-of-freedom (2dof), and multiple-degree-of-freedom (mdof) mechanical systems are considered. Viscoelastic models involving fractional derivatives instead of common derivatives: the generalized Maxwell model with one or two fractional parameters (orders of fractional derivatives), the generalized Kelvin–Voigt model and the generalized standard linear solid model have been used. The Laplace integral transform method is employed as a method of solution. However, as distinct to the traditional approach, when rationalization of a characteristic equation with fractional powers is carried out during the transition from image to pre-image, here for every problem a nonrationalized characteristic equation is solved by the method suggested by the authors. As a result of such an approach, the solution is obtained in the form of the sum of two terms, one of which governs the drift of the system's equilibrium position and is defined by the quasi-static processes of creep and relaxation occurring in the system, and the other term describes damped vibrations around the equilibrium position and is determined by the systems's inertia and energy dissipation.