A finite element approach for determining the dynamic behaviours of multiscale beams using the semi-continuum model

Abstract

The dynamic mechanical behaviours of a multiscale beam with nanoscaled thickness are investigated. The thickness of the multiscale beam is divided to some discrete atomic layers, while the scales of length and width are within the classical continuum framework. Such a semi-continuum model is employed to measure the size dependence of the multiscale beam. Based on the governing equation of motion, the finite element expressions for transverse vibration are achieved through the weighted residual method. By using the Hermitian interpolation function, the structural mass matrix and stiffness matrix are obtained, and the system equation via finite element approach is got as well. Subsequently, the finite element analysis is carried out with one hundred equal length units. The results show that the natural frequencies of the multiscale beam are related to the relaxation coefficient of surface atoms. When the relaxation coefficient is more than 1, the fundamental frequency parameter is positively correlated with the number of atomic layers, while when the value is less than 1, it becomes negatively correlated. The lateral displacement and rotational angle of the first two modal functions are determined, and it is indicated that the phase of the angular variation is always half a period faster than the phase of the deflection variation.