Data-Driven and Analytical Investigation of Free Vibration in Agglomerated FG-CNT Coated Nanoshells

This study presents a unified analytical and data-driven framework for predicting free vibration frequencies of coated functionally graded carbon nanotube-reinforced (FG-CNT) nanoshells. This study examines two coated FG schemes (Hardcore and Softcore) combined with multiple material distribution patterns. A comprehensive analytical database is generated using higher-order shear deformation theory (HSDT) and the Galerkin method, incorporating nonlocal strain-gradient (NSGT) size effects and accounting for CNT agglomeration through a two-parameter homogenization model. The governing equations are derived via Hamilton’s principle and solved analytically for shells resting on Winkler–Pasternak elastic foundations under various boundary conditions. A Random Forest regression surrogate model is trained on this analytical database, delivering exceptional prediction accuracy with computational efficiency dramatically improved relative to repeated analytical evaluation. SHAP (Shapley Additive Explanations) analysis identifies the dominant parameters governing vibration response and preserves physical interpretability. Parametric studies reveal that CNT agglomeration significantly degrades effective stiffness, while material grading indices enhance Hardcore shell rigidity but reduce Softcore shell stiffness. The proposed framework enables rapid frequency prediction for design optimization and structural health monitoring applications without repeatedly solving governing equations.