Exact closed form solution for the analysis of the transverse vibration mode of a Nano-Timoshenko beam with multiple concentrated masses

Abstract

Concentrated masses mounted on a Nanobeam have important effects on its transverse vibrations. In this paper the effects of concentrated masses on lateral vibration of a Nanobeam is investigated by utilizing two different points of view about shear force in the beam. The investigation is conducted with and without considering small scale effect on shear force. Timoshenko beam and nonlocal theories have been used in Newton's second law to model transverse dynamic behavior of a Nanobeam; then the mathematical model of concentrated masses is imposed into the equations of motion by using Dirac's delta function. After obtaining exact closed form solution, basic functions are used to simplify calculations. Different parameters of the Nanobeam are used to study transverse vibration of Nanobeam carrying concentrated masses. These parameters include number, mass and location of concentrated masses and length, cross section area and small scale factor of Nanobeam. Finally, the effects of these parameters are presented in diagrams for two points of view. The diagrams show significant difference between consideration and disregarding of small scale factor in shear force in higher natural frequencies of Nanobeam.