Size-dependent free vibration analysis of nonlocal Timoshenko nanobeams on a Winkler foundation using the initial value method

Accurate modeling of nanoscale beam-like structures requires the consideration of size-dependent effects that cannot be captured by classical continuum theories. In this study, the free vibration behavior of nonlocal Timoshenko nanobeams resting on a Winkler-type elastic foundation is investigated within the framework of Eringen’s nonlocal elasticity theory. Obtaining exact or closed-form solutions becomes increasingly difficult when nonlocal effects, Timoshenko beam kinematics, elastic foundation interaction, and various classical boundary conditions are considered simultaneously. To address this, a unified semi-analytical solution framework based on the initial value method (IVM) and the approximate transfer matrix (ATM) approach is employed to compute the dimensionless natural frequencies of the system. Unlike existing discretization-based approaches, the proposed framework enables the direct incorporation of various classical boundary conditions without requiring global matrix assembly, offering a systematic and computationally efficient alternative for parametric analyses of nonlocal beam–foundation systems. The accuracy, convergence, and numerical stability of the formulation are verified through comparisons with available results in the literature. A detailed parametric study is carried out to examine the effects of the nonlocal parameter, Winkler foundation stiffness, geometric aspect ratio, boundary conditions, and vibration modes on the vibrational response of the nanobeams. The results indicate that increasing the elastic foundation stiffness leads to higher natural frequencies, whereas stronger nonlocal effects reduce them. The proposed framework provides an efficient and reliable basis for the vibration analysis and parametric assessment of nonlocal nanobeam–foundation systems.