Forced oscillations of a body attached to a viscoelastic rod of fractional derivative type

We study forced oscillations of a rod with a body attached to its free end so that the motion of a system is described by two sets of equations, one of integer and the other of the fractional order. To the constitutive equation we associate a single function of complex variable that plays a key role in finding the solution of the system and in determining its properties. This function could be defined for a linear viscoelastic bodies of integer/fractional derivative type

On a system of equations arising in viscoelasticity theory of fractional type

We study a system of partial differential equations with integer and fractional derivatives arising in the study of forced oscillatory motion of a viscoelastic rod. We propose a new approach considering a quotient of relations appearing in the constitutive equation instead the constitutive equation itself. Both, a rod and a body are assumed to have finite mass. The motion of a body is assumed to be translatory. Existence and uniqueness for the corresponding initial-boundary value problem is proved within the spaces of functions and distributions

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of viscoelasticity and in the system identfica- tion theory. Also, the existence and uniqueness of a solution to a general linear fractional differential equation in the space of tempered distributions is given

Nano and viscoelastic Beck's column on elastic foundation

Beck's type column on Winkler type foundation is the subject of the present analysis. Instead of the Bernoulli-Euler model describing the rod, two generalized models will be adopted: Eringen non-local model corresponding to nano-rods and viscoelastic model of fractional Kelvin-Voigt type. The analysis shows that for nano-rod, the Herrmann-Smith paradox holds while for viscoelastic rod it does not

Space-time fractional Zener wave equation

Space-time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analyzed. This model includes waves with finite speed, as well as non-propagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy problem are proved. Special cases are investigated and numerical examples are presented

On a initial value problem arising in mechanics

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

Distributional framework for solving fractional differential equations

We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line