On a model of viscoelastic rod in unilateral contact with a rigid wall

Abstract

We study translatory motion of a body to which a viscoelastic rod with the constitutive equation with fractional derivatives is attached. The body with a rod impacts against a rigid wall. It is shown that the problem is described with a coupled system of differential equations having integer and fractional derivatives having the form x (2) = - f; f + af (a) = x + bx (a) , x(0) = 0, x (1) (0) = 1. The unique solvability in S′ + is proved and interpretation of solutions is given. Also, some a priori estimates of the solution are given. In particular, we showed that restrictions on coefficients that follow from the second law of thermodynamics imply that the velocity after the impact is smaller than the velocity before the impact. © The Author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.