Abstract
In this paper, linear vibrations of axially moving systems which are modelled by a fractional derivative are considered. The approximate analytical solution is obtained by applying the method of multiple scales. Including stability analysis, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted by graphs. It is determined that the external excitation force acting on the system has an effect on the stiffness of the system. Moreover, the general algorithm developed can be applied to many problems for linear vibrations of continuum.
Highlights
Fractional derivatives are useful for describing the occurrence of vibrations in engineering practice
The general model proposed for continuum is linear and one-dimensional
The effect of the damping term which is obtained from viscoelastic material properties is modelled with a fractional derivative
Summary
Fractional derivatives are useful for describing the occurrence of vibrations in engineering practice. The studies involving fractional calculus and its applications to mechanical problems appear widely in different studies [ ]. The advances in fractional calculus focus on modern examples in differential and integral equations, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology and electrochemistry [ ]. Many different linear or nonlinear models addressing vibrations of continuum appear in the literature. Some of these works are as follows: Pakdemirli [ ] developed a general operator technique to analyse the vibrations of a continuous system with an arbitrary number of coupled differential equations. Özhan and Pakdemirli [ – ] suggested the general solution procedure to investigate a more general class of continuous systems such as gyroscopic and viscoelastic systems. A general solution is adapted to solve the dynamic problems constituting continuum
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