This study explores the dynamics of functionally graded material (FGM) rotors using a finite element approach based on Timoshenko beam theory. The FGM rotor, composed of SUS304/silicon nitride, is analyzed for critical speeds and stability, incorporating gyroscopic effects, rotary inertia, and shear deformation. Results show that material gradation (power-law index k) significantly impacts natural frequencies, with ceramic-rich compositions achieving up to 15% higher critical speeds. Campbell diagrams reveal forward and backward whirl modes, identifying critical speeds at 1460 rpm, 1980 rpm, and 2650 rpm. The study highlights the inverse relationship between slenderness ratio (L/d) and natural frequencies, stabilizing beyond L/d>10. Validated with benchmark solutions ( 2%), the model proves effective for FGM rotor design in high-performance turbomachinery.