Large deflection model for nonlinear modal dynamics in functionally graded multi-thick-disk shaft with gradation variations and porosity imperfections

This study presents an in-depth investigation into the nonlinear free vibration behavior of highly deformable multi-disk shaft systems composed of functionally graded materials with porosity imperfection. The analysis incorporates rotary inertia, gyroscopic coupling, disk thickness, and axial restraint effects to accurately capture the system’s complex dynamics. Closed-form expressions for both linear and nonlinear resonance frequencies are derived using the method of multiple scales and validated through finite element method simulations and numerical analyses. Time histories, Campbell diagrams, fast Fourier transforms, and Poincaré maps highlight the significant influence of material gradation and porosity on dynamic behavior. Nonlinear resonance frequencies exceed their linear counterparts and are highly sensitive to initial conditions, porosity levels, and multiporous functionally graded disks. Variations in gradient indices and porosity imperfections significantly affect stiffness, mass distribution, and modal parameters. Increasing the grading index lowers nonlinear resonance frequencies, with non-uniform grading accelerating this shift, while radial and localized porosity imperfections further reduce resonance frequencies and alter their trends. Replacing a thin disk with a thick disk significantly modifies system stiffness and modal properties, strengthening gyroscopic effects and lowering critical speeds. These findings emphasize the need for accurate nonlinear modeling of a shaft with multiple rigid disks, along with material gradation and porosity imperfections, to improve the dynamic performance and reliability of high-speed rotating machinery.