Zepeda Aguilar, Mario

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Vibration of cylinders of fractal structures
(2016-06) Zepeda Aguilar, Mario; Steinberg, Lev; College of Arts and Sciences - Sciences; Cruz, Angel; Castillo, Paul; Department of Mathematics; Alers Valentin, Hilton
The objective of this study was to develop a dynamical model of elastic fractal solids. This model is built on the concepts of the fractal continuum, fractal time, and the corresponding fractal derivatives. Also this approach incorporates a modification of basic laws of continuum mechanics such as the conservation of mass, the balance of momentum, and the constitutive relationships. We established a corresponding linearized model to study the elastic vibrations. We considered special cases of torsional, longitudinal and transverse vibrations of fractal circular cylinders. We introduced fractal cylindrical coordinates and obtained systems of fractal partial differential equations in these coordinates for each of the mentioned cases. For each case the obtained corresponding analytic solutions exhibit distinctive size and time effect properties. In particular, we show the dependence of resonance frequencies and resonance forms on the internal fractal structures and the size of the cylinders.