Analyzing torsional vibration in restrained functionally graded nanobeams: nonlocal Lam strain gradient approach

Abstract

This research explores the free torsional vibration behavior of a functionally graded (FG) circular nanobeam, constrained at both ends by torsion springs, focusing on the size effect. To account for this size effect in the FG nanobeam (FGNB), the nonlocal Lam strain gradient theory is employed, which incorporates three-dimensional scale parameters alongside classical material parameters. The composition of the FGNB, consisting of ceramic and metal components, follows the power law rule, allowing for a gradation in material distribution. The primary aim of this study is to develop a model for the FGNB that includes torsion springs at its boundaries, thereby providing a comprehensive solution applicable to general boundary conditions. This model accommodates general boundary conditions, making it versatile for various applications. The study utilizes the Fourier sine series and Stokes' transform to achieve this goal. These analytical techniques enable the presentation of solutions that can accurately investigate the torsional dynamics of the FGNB under the influence of size effects and boundary constraints. Results from this analysis reveal that the torsional vibration response of the FGNB is significantly influenced by the size effect, with notable variations in vibration behavior under different boundary stiffness conditions. Including torsion springs at the boundaries introduces a dynamic aspect to the system, allowing the model to apply to real-world applications more accurately. The findings demonstrate that the nonlocal and strain gradient parameters critically affect the torsional vibration frequencies.