On the Magnetically Tunable Free Damped-Vibration of L-Shaped Composite Spherical Panels Made of GPL-Reinforced Magnetorheological Elastomers: An Element-Based GDQ Approach

In practical engineering applications, curved structures rarely conform to idealized rectangular or circular planforms and often involve far more intricate geometries. Among these, L-shaped spherical panels have emerged as a structurally significant form, found in subsystem interfaces, aerospace fuselage junctions, complex biomedical shells, and multifunctional architectural surfaces. This study explores the free damped-vibration behavior of such panels constructed from a graphene platelet (GPL)-reinforced magnetorheological elastomer (MRE) nanocomposite. Unlike conventional elastic matrices, the MRE base material exhibits time- and field-dependent viscoelastic behavior, influenced by both magnetic field intensity and ferromagnetic content. This behavior is mathematically formulated through an experimentally validated generalized Kelvin–Voigt-type model, tailored to represent the storage and dissipation characteristics of the matrix under dynamic excitation. The reinforcing particles are graded through the panel thickness. The effective elastic properties of the composite are homogenized using the Halpin–Tsai micromechanical model, accounting for the influence of GPL content and sizes. To address the geometric complexity, a hybrid element-based GDQ (generalized differential quadrature) approach is developed. The L-shaped spherical panel is subdivided into rectangular elements, each governed by equations derived using Hamilton’s principle, first-order shear deformation theory, and Sander’s strain-displacement relations. Discretization via quadrature nodes enables the GDQ method to transform the governing PDEs into an efficient algebraic system. The global system is constructed by enforcing both displacement and force continuity at shared nodes and applying appropriate boundary conditions. The proposed framework achieves excellent accuracy in capturing frequencies and loss factors, demonstrating its capability for efficient dynamic analysis of non-standard. In addition to validating the accuracy of the proposed approach against benchmark problems, the study reveals distinct mode-switching and mode-jumping phenomena triggered by changes in geometric parameters—highlighting the sensitivity of vibrational behavior to panel shape and reinforcing the need for precise modeling in advanced smart structures.