Free vibration analysis on fractal beams

In this work, a generalized form of motion’s equations for free vibration employing the Balankin’s fractal derivatives ($F^{\alpha }$-derivatives) in the fractal continuum framework is suggested. Interrelation between $F^{\alpha }$ and ordinary derivatives makes possible to transform the vector differential operators in the fractal domain ${\Re }_x^3$ of vector differential calculus into the corresponding fractal continuum ${\Re }_{\xi }^3$, so the fractal free vibration equation for self-similar beams is derived. The solution of the proposed fractal equation is obtained, and several practical examples involving beams with classical boundary conditions are solved to discuss the structural implications.