Coupling Effects of Strain Gradient and Surface Energy on Dynamic Instability of Small-Scale Timoshenko Beams Traversed by Sequential Moving Nanoparticles

Abstract

This study aims to explore the dynamic instability of micro and nano-sized Timoshenko beams as they are traversed by sequentially moving nanoparticles. The beams, characterized by a rectangular cross-section and homogeneity, are situated within a Pasternak foundation, which provides a supportive elastic medium. The research investigation determines nanoparticle inertia effects at velocity while establishing motion equations through Hamilton's principle. The model unites nonlinear von Kàrmàn strain-displacement kinematics with strain gradient theory and Gurtin-Murdoch small-scale accounting. The system's behavior gets analyzed through the implementation of Galerkin method which derives time-periodic motion equations. The incremental harmonic balance approach develops stability boundary maps that separate stable and unstable regions through which analysts can examine parameter spaces containing moving particle mass and velocity values. This study evaluates how different parameters like beam diameters together with small-scale characteristics and elastic medium constants and residual stress and axial compressive forces affect the stability diagram. The analysis demonstrates that stability parameters become substantially modified when researchers include length scale characteristics along with surface effects. The outcome reveals that axial compressive forces reduce stability yet environmental effects strengthen the stability of small-scale beams which leads to transition curve movements towards faster moving particles velocities. This study contributes fundamental knowledge about dynamic instability effects in small-scale beams which will help future advances in nanotechnology and materials science. © 2025 The Authors