A computationally efficient semi-analytical method for size-dependent free vibration of nonlocal timoshenko nanobeams

This study presents a semi-analytical framework for the size-dependent free vibration analysis of Timoshenko nanobeams based on nonlocal elasticity theory. The Initial Value Method (IVM) combined with the Approximate Transfer Matrix (ATM) approach is employed to compute natural frequencies. The proposed method captures nanoscale effects with high accuracy and reduced computational effort, eliminating the need for closed-form symbolic expressions and offering a practical alternative to conventional numerical techniques. Convergence analyses and comparison with a benchmark solution confirm the robustness and validity of the solution method. Parametric studies reveal that increasing μ reduces the natural frequencies in supported, clamped–simply supported, and clamped–clamped beams, demonstrating the expected softening behavior due to nonlocal effects. For the clamped–free beam, an increase in μ leads to a hardening response for the fundamental mode, while higher modes retain the classical softening trend. Additionally, increasing L/h results in higher frequencies, and the influence of nonlocality becomes more pronounced for higher-order modes. The ATM–IVM framework provides a robust and efficient tool for modeling, designing, and optimizing advanced nanostructured materials, where an accurate representation of size effects is crucial.