Size-dependent free vibration analysis of Rayleigh nanobeams based on the stress-driven nonlocal model using Fourier series and Stokes’ transformation

Abstract

In this study, the size-dependent free vibration behavior of Rayleigh nanobeams has been investigated using Fourier series and the Stokes transform within the framework of a stress-driven nonlocal model (SDNM). In the developed solution approach, the displacement field is expressed as a Fourier sine series, and higher-order derivatives are obtained using the Stokes transform, thereby formulating a single eigenvalue problem that simultaneously incorporates the nonlocal parameter, rotational inertia, and boundary conditions (BCs). The results obtained have revealed that an increase in the stress-based nonlocal parameter causes a significant increase in frequencies, thereby creating a stiffening effect on the system behavior. Furthermore, the increase in the number of modes has made the effect of rotational inertia more pronounced due to the formation of multiple bending regions along the beam, highlighting the importance of Rayleigh beam theory (RBT), particularly in modeling higher modes. Additionally, it has been determined that the stiffness of the BCs has a significant effect on the frequencies, with stiffer support conditions producing higher frequency values. The original contribution of this study is the integration of the stress-based nonlocal RBT with Fourier series and the Stokes transform, and the presentation of a high-accuracy solution framework under various BCs. It is anticipated that the results obtained will contribute to a better understanding of the role of scale effects and BCs on the dynamic behavior of nanobeams.