Construction of fractal stiffness and mass matrices for two-dimensional self-similar frames

In this work, we constructed the dynamics equilibrium equation for fractal manifolds based on the concept of fractal calculus introduced recently by Balankin, using their fractal continuum derivatives, as an extension of traditional differential calculus to spaces and functions characterized by fractal geometry.

The generalized form of the equation of motion for two-dimensional self-similar frames subjected to forced vibration incorporates both the stiffness and mass matrices, which account for the hierarchical structure and scaling properties of the frame's geometry; allowing the accurate modeling of dynamic responses by considering the influence of fractal continuity and irregular distribution of mass and rigidity, reflecting the distinctive physical behavior of such frames under external influence.

The fundamental interrelation between Balankin fractal derivatives and ordinary derivatives establishes a connection that enables the transformation of vector differential operators defined in the fractal domain R_x³ into their counterparts within the fractal continuum R_ξ³. This relationship behaves as a mathematical bridge, mapping complex fractal structures (characterized by non-integer dimensions and self-similarity) into a generalized fractal continuum framework.

The alpha and beta parameters of order α, β ∈ (0,1] are added to the new fractal matrices via this transformation, preserving essential fractal properties (geometry and topology) and providing new analytical tools and models for physical phenomena occurring within fractal continua. The influence of these parameters on the vibrational characteristics of the frames is analyzed graphically. Some mechanical implications are also discussed.