This study explores the use of fractional partial differential equations to model the wave propagation through a one-dimensional complex and heterogeneous medium. In particular, this work discusses the use of fractional calculus to obtain closed form analytical solutions for the dispersion and propagation of elastic waves in a periodic, bi-material rod. From a mathematical standpoint, the approach allows converting a partial differential equation having spatially variable coefficients (i.e. the traditional wave equation in periodic media) to a space-fractional wave equation with constant coefficients. We show that the equivalent fractional equation exhibits a frequency-dependent and complex fractional order. Although this conversion might appear to increase the overall complexity of the model, in practice it enables obtaining closed form analytical solutions of wave propagation problems through inhomogeneous media.
The analytical solution to the space-fractional equation is obtained for the steady state response under harmonic loading. The result is compared to the traditional finite element solution of the wave equation in periodic media showing that the new approach is able to provide a reliable and accurate analytical representation of the dynamic response of the medium.
This study explores the use of fractional differential equations to model the vibration of single (SDOF) and multiple degree of freedom (MDOF) discrete parameter systems. In particular, we explore methodologies to simulate the dynamic response of discrete systems having non-uniform coefficients (that is, distribution of mass, damping, and stiffness) by using fractional order models. Transfer functions are used to convert a traditional integer order model into a fractional order model able to match, often times exactly, the dynamic response of the active degree in the initial integer order system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting differential models have both frequency-dependent and complex fractional order. The presented methodology is practically equivalent to a model order reduction technique that is able to match the response of non-uniform MDOF systems to a simple fractional single degree of freedom (F-SDOF) systems. The implications of this type of modeling approach will be discussed.
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