On a initial value problem arising in mechanics

TL;DR

This paper investigates an initial value problem involving fractional differential equations modeling forced oscillations in viscoelastic rods, providing explicit solutions and analyzing thermodynamic restrictions.

Contribution

It offers explicit convolution integral solutions for a class of linear viscoelastic solids and unifies previous results under a common framework.

Findings

Explicit solution formulas derived for linear viscoelastic models

Thermodynamic restrictions shape the solution structure

Previous results are special cases of the new analysis

Abstract

We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of a solution for a class of linear viscoelastic solids is given in terms of a convolution integral. Restrictions on storage and loss moduli following from the Second Law of Thermodynamics play the crucial role in establishing the form of the solution. Some previous results are shown to be special cases of the present analysis.