Damped transverse vibrations of magnetostrictive functionally graded (FGMT) beams on viscoelastic foundations: A coupled FEM-Galerkin-ANN framework

The damped transverse vibrations of magnetostrictive functionally graded (FGMT) beams incorporating a Terfenol-D layer and resting on a Kelvin–Voigt viscoelastic foundation are investigated in this study. The mechanical model accounts for material gradation, magnetostrictive damping, and both elastic and viscous foundation effects. It is formulated within the framework of Euler–Bernoulli beam theory and derived using Hamilton’s principle. To determine the damped natural frequencies, mode shapes, and transient responses under various boundary conditions, a finite element model (FEM) is developed and employed as the primary numerical tool. For semi-analytical validation, a Galerkin-based formulation is also implemented. Furthermore, FEM-generated data are used to train an artificial neural network (ANN) surrogate model, enabling rapid and accurate prediction of the vibration characteristics over a wide range of physical parameters. The results demonstrate that the dynamic behavior of the beam is strongly influenced by the boundary conditions, material gradation, foundation stiffness, and viscoelastic damping. Excellent agreement is observed among the FEM, Galerkin, and ANN results, with very low relative errors. Consequently, the proposed coupled FEM–Galerkin–ANN framework provides an efficient and reliable tool for the analysis, design, and optimization of intelligent FGMT structures resting on viscoelastic foundations.