abstract
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In this work, a generalized form of motion's equations for free vibration employing the Balankin's fractal derivatives (F¿-derivatives) in the fractal continuum framework is suggested. Interrelation between F¿ and ordinary derivatives makes possible to transform the vector differential operators in the fractal domain ¿
x 3 of vector differential calculus into the corresponding fractal continuum ¿¿ 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. © 2025 Elsevier Masson SAS