Moving point load on a beam with viscoelastic foundation containing fractional derivatives of complex order

Abstract

In this paper, the dynamical behavior of the Euler-Bernoulli beam resting on a generalized Kelvin-Voigt-type viscoelastic foundation, subjected to a moving point load, is analyzed. Generalization is done in the sense of fractional derivatives of complex-order type. Mixed initial-boundary value problem is formulated, and the solution is given in the form of Fourier series with respect to space variable, where coefficients satisfy a certain system of ordinary fractional differential equations of complex fractional order with respect to time variable. Thermodynamical restrictions on the parameters of the model are also given. It is shown that those are sufficient for the existence and the uniqueness of the solution. The solution of the problem is expressed in closed form, by using the inverse Laplace transform method. A numerical example confirming the invoked theory is presented.