Fractal-tropical idempotent matrices for nonlinear vibration control of fractal MEMS

The present paper proposes a theoretical framework for the nonlinear vibration control of fractal micro-electromechanical systems (MEMS) based on fractal-tropical idempotent matrices. The present study extends the theory of fifth-order fractal-tropical matrix idempotents, thereby establishing the mathematical foundations for the application of these algebraic structures to vibration control in fractal MEMS. The two-scale fractal dimension is utilized as a rigorous geometric descriptor, thereby linking MEMS geometry and algebraic structure. The development of the theoretical formulation of modal preservation filters is achieved through the utilisation of fractal-tropical idempotent matrices. The demonstration is provided that the idempotence condition ensures operational stability and preserves self-similar features. The formulation of optimal control problems with idempotent constraints is undertaken, and it is verified that constraint matrices satisfying idempotence guarantee consistency in iterative solution procedures. A novel application of pull-in analysis demonstrates that the critical pull-in voltage is directly modulated by the two-scale fractal dimension, with more irregular fractal structures requiring higher critical pull-in voltages. Theoretical analysis indicates that the twoscale fractal dimension exerts a direct regulatory influence on the coupling structure of control systems. Specifically, a reduced fractal dimension engenders enhanced neighborhood correlation in control matrices, thereby furnishing a mathematically sound foundation for the adaptation of control strategies to fractal geometries. The present work establishes a direct theoretical bridge between fractal geometry and nonlinear vibration control, thus offering a new algebraic tool for the analysis and design of fractal MEMS devices.