Analysis of the effect of nonlocal factors on the vibration characteristics of Euler–Bernoulli nonlocal nanobeams on deformed foundations

The classical Euler–Bernoulli beam theory, rooted in continuum mechanics and localization theories, fails to incorporate the effects of foundation deformation and atomic lattice interactions, thus inadequately capturing the true mechanical properties of nanobeams. This study aims to address these limitations by proposing a novel computational framework to accurately model the global coupling and mechanical behavior of nanobeams. First, the computational approach integrates both foundation deformation and atomic lattice interactions, constructing a physical model for the nonlocal vibration of nanobeams under arbitrary loading conditions. A validation methodology is also developed to ensure the model’s reliability. Second, using the Laplace transform, the time-domain problem is transformed into a frequency-domain analysis. The Hasselman complex modal synthesis method is employed, and for the first time, the space-state transfer function of the nanobeam vibration model, based on a modified Euler–Bernoulli beam theory incorporating nonlocal effects, is derived. Analytical solutions are presented and validated. Finally, the mechanism of global coupling in nanobeams is examined through the nonlocal intrinsic structure, using the material points of the beam to accurately reflect nanoscale mechanical behavior. Results reveal that the nonlocal factor (ranging from 0 to 0.3) significantly influences the frequency peaks of the nanobeam. As the mode order nnn increases, the frequency peak shifts along the transverse axis toward decreasing nonlocal factors and its magnitude diminishes. Conversely, with increasing beam length, the frequency peak moves toward increasing nonlocal factors on the transverse axis, and the magnitude increases. Furthermore, the influence of the nonlocal factor on the frequency and amplitude is pronounced in lower-order modes, gradually diminishing as the mode order increases. These findings not only enhance the understanding of the vibration characteristics of nanobeams but also provide a robust framework for analyzing the impact of nonlocal effects, offering new insights and avenues for future research in nanostructural mechanics.