Non-linear problems of fractional calculus in modeling of mechanical systems

The paper presents mathematical formulation and numerical algorithm for solving non-linear fractional-order differential equations (FDEs) modeling mechanical systems. The method presented in the paper involves the notion of variational inequalities. It is applied to one-term FDEs which are linear with respect to the fractional derivative. The examples of rheological models containing, in addition to fractional elements, the non-linear elastic, viscous and plastic elements are presented. The difference time-discretization schemes of Euler and Runge–Kutta types for solving initial-value problems are proposed. A special attention is paid to analysis of an original elastic-visco-plastic fractional model of asphalt–aggregate mixes being a modification of the classical Huet–Sayegh model. The results of numerical simulations of mechanical systems subjected to harmonic kinematic excitations are presented.

► We analyzed the mathematical formulation of non-linear fractional-order differential equations modeling mechanical systems.
► We proposed time-discretization schemes of Euler and Runge–Kutta types for solving initial-value problems.
► The method can be applied for mechanical systems containing non-linear elastic and dissipative components.