Fractional Approach to Dynamic Analysis of Coupled Circular Plates with Hereditary Effects

This study investigates the dynamic behaviour of coupled circular plates connected by viscoelastic layers exhibiting creep effects. Such multilayer systems are widely used in aerospace, civil engineering, and microsystems, where time-dependent material properties and long-term loading significantly affect structural stability and performance.

The proposed analytical framework applies fractional calculus to capture hereditary effects in viscoelastic materials, offering a more accurate representation of time-dependent behaviour than classical models. Starting from D’Alembert’s principle and the theory of hereditary viscoelasticity, we derive a system of partial integro-differential equations describing transverse oscillations of the plates. These equations are systematically reduced to ordinary integro-differential form using Bernoulli’s method and solved via Laplace transforms, enabling closed-form solutions for key dynamic characteristics.

The results provide explicit expressions for natural frequencies, mode shapes, and time-dependent response functions. Analysis reveals the emergence of multi-frequency vibration regimes, with modal families remaining temporally uncoupled. This property facilitates precise identification of resonance conditions and dynamic absorption phenomena. The fractional parameter acts as a tunable damping factor: lower values allow prolonged oscillations, while higher values induce rapid decay. Additionally, increasing the kinetic stiffness of coupling layers raises vibration frequencies and amplifies hereditary effects.

These findings deliver a robust tool for designing multilayered structures with tailored dynamic responses. By adjusting fractional order and stiffness ratios, engineers can optimise vibration control, enhance stability, and mitigate long-term deformation risks under complex loading conditions.