Interaction of the fundamental frequencies of a torsional cantilever nanobeam and spring mass system single degree of freedom (SDOF) under axial load, including buckling

Abstract

In this study we present the interactions of the fundamental frequencies of a nanomanufacturing coupled system by exploring the natural frequencies of the subsystems. These nanomanufacturing subsystems function in concert, e.g., a cantilever beam with spring-mass. The individual subsystems are studied under free vibration to generate the natural and buckling frequencies. These subsystems, when under free vibration, generate unique local natural frequencies that interact to form a unique global natural frequency. This allows for greater control and improved sensitivity for scanning and shaping nano surfaces, by allowing selective variation of the local frequency of one system to influence the global system frequencies. In this investigation, a nanobeam with arbitrary boundary conditions is used to model the system and the effects on the parameters of interest are studied. Euler–Bernoulli theory is applied in conjunction with Eringen’s theory of nonlocal continuum theory to model the small-scale effects due to the size of the beam under consideration. The coupled equations are solved using separation of variables for the local and global frequencies. The nanobeam is restrained with an adjustable torsional spring and pin at one end. The boundary condition at the free end is a spring-mass system with axial load. Altering the torsional, transverse spring stiffness and mass increases or decreases the natural frequencies. The motions of the beam and the tip-mass generates a frequency response during contact interactions. The tip response frequency is used to determine the maximum displacements (penetration depth) and accelerations (contact forces) in a sample during nanomanufacturing.