Nonlocal normal modes in nanoscale dynamical systems

Abstract

This paper introduces the idea of nonlocal normal modes arising in the dynamic analysis of nanoscale structures. A nonlocal finite element approach is developed for the axial vibration of nanorods, bending vibration of nanobeams and transverse vibration of nanoplates. Explicit expressions of the element mass and stiffness matrices are derived in closed-form as functions of a length-scale parameter. In general the mass matrix can be expressed as a sum of the classical local mass matrix and a nonlocal part. The nonlocal part of the mass matrix is scale-dependent and vanishes for systems with larger lengths. Classical modal analysis and perturbation method are used to understand the dynamic behaviour of discrete nonlocal systems in the light of classical local systems. The conditions for the existence of classical normal modes for undamped and damped nonlocal systems are established. Closed-form approximate expressions of nonlocal natural frequencies, modes and frequency response functions are derived. Results derived in the paper are illustrated using examples of axial and bending vibration of nanotubes and transverse vibration of graphene sheets.