Dynamic Analysis of Functionally Graded Nanobeams on Viscoelastic Foundation Using a Fourier Series Based Semi-Analytical Method

Kadioglu H. G., YAYLI M. Ö.

JOURNAL OF VIBRATION ENGINEERING & TECHNOLOGIES, cilt.14, sa.8, 2026 (SCI-Expanded, Scopus)

Özet

Purpose: This study investigates the free vibration behavior of functionally graded (FG) nanobeams resting on a Kelvin-Voigt viscoelastic foundation with elastically restrained ends, using the stress-driven nonlocal model to capture size effects without the paradoxes of strain-driven formulations.Methods: The formulation is based on Euler-Bernoulli beam theory combined with stress-driven nonlocal elasticity, with material properties varying through the thickness according to a power law. The displacement field is expressed as a Fourier sine series, and Stokes' transformations are used to obtain its higher-order derivatives. Arbitrary combinations of classical and non-classical boundary conditions are thereby reduced to a single nonlinear complex characteristic equation. An iterative root-search procedure, extended to complex-valued functions, is used to solve it. The convergence of the method is examined, and the results are validated against available local and nonlocal solutions in the literature.Results: Relative errors remain below 0.02% for nonlocal benchmarks and within 0.61% for FG benchmarks. Increasing the scale parameter raises the dimensionless frequencies through a stiffening effect, and this effect becomes more pronounced at higher vibration modes. Its relative influence decreases as foundation stiffness increases, since the foundation progressively dominates the system stiffness. In contrast, it increases with foundation damping. The damping component of the complex frequency is independent of boundary conditions, scale parameter, and mode number. It is governed by the effective density and therefore increases with the power-law exponent. Foundation stiffness has its strongest effect on the clamped-free configuration.Results: The proposed approach provides an efficient and unified framework for the dynamic analysis of FG nanostructures interacting with viscoelastic media and can support the design of nanoscale devices such as sensors and resonators.